Method for calculating Control Attainable Set of redundant drive system under linear constraint

Abstract

Provided is a method for calculating a control attainable set of a redundant drive system under a linear constraint, which relates to the technical field of dynamics control allocation of the redundant drive system. The method first constructs a control attainable set problem for a redundant drive system under each pair of linear constraint control components, and then classifies boundary surfaces corresponding to a control set into three types of rectangular boundary surfaces and one type of triangular boundary surface. By grouping the boundary surfaces, the method determines a key boundary surface for each group to form a boundary surface set. After removing duplicate boundary surfaces from the boundary surface set, the method calculates a boundary surface of a control attainable set, and finally obtains the control attainable set.

Description

CROSS REFERENCE TO RELATED APPLICATION

This patent application claims the benefit and priority of Chinese Patent Application No. 202310433407.2, filed with the China National Intellectual Property Administration on Apr. 21, 2023, the disclosure of which is incorporated by reference herein in its entirety as part of the present application.

TECHNICAL FIELD

The present disclosure relates to the technical field of dynamics control allocation of a redundant drive system, and in particular, to a method for calculating a control attainable set of a redundant drive system under a linear constraint.

BACKGROUND

A control attainable set of a redundant drive system can quantitatively characterize a control capability of the system, and calculation of the control attainable set is an inverse problem of control allocation. The control allocation is responsible for allocating an expected system control vector to each redundant actuator for execution. The calculation of the control attainable set is to determine, in the case of a known variation range of each actuator, a boundary of a system control attainable vector that can be achieved by all actuators acting simultaneously, to obtain the control capability of the redundant drive system, especially a control capability of the system after some actuators fail. A control allocation method based on the control attainable set has become a hot research topic in the field of control allocation.

A control attainable set of a parallel configuration redundant drive system can be mathematically expressed as follows:

$\begin{array}{cc}\Phi =\left\{v❘v=B·u,u\in \Omega \right\}& \left(1-1\right)\end{array}$

• • In the above expression, u represents a control vector, and u=(u₁, . . . , u_m)^T, which represents a control input of the redundant drive system, where T represents a matrix transpose symbol, and an i^th control component is a control action of a corresponding i^th actuator, where 1≤i≤m, m represents a quantity of actuators, u_imin≤u_i≤u_imax≤u_imin represents a minimum constraint value of the control action of the i^th actuator, and u_imax represents a maximum constraint value of the control action of the i^th actuator; there is often a linear or nonlinear constraint between each u_i; Ω represents a control set, and Ω={u}; v represents a control attainable vector of the redundant drive system, and v=(v₁, L, v_n)^T, which represents a control output of the redundant drive system, where v_j represents a j^th control attainable component, 1≤j≤n, n represents a dimension of the control attainable vector, and n<m; Φ represents the control attainable set; and B represents an n-row and m-column control efficiency matrix.

The above expression (1-1) is physically used to describe how to determine, based on the control efficiency matrix B, the control attainable vector set Φ composed of n control outputs of the redundant drive system, provided that the control vector set Ω composed of m control inputs of the redundant drive system is known. For example, a physical meaning of a control attainable set of a four-wheel independently driven and independently steered vehicle is as follows:

• • 1) Four longitudinal forces of four wheels are F_L1, F_L2, F_L3, and F_L4 respectively, and four lateral forces of the four wheels are, F_T1, F_T2, F_T3 and F_T4 respectively.
  • 2) A control vector u_v=(F_L1, F_T1, F_L2, F_T2, F_L3, F_T3, F_L4, F_T4)^T containing eight control components is set for the four-wheel independently driven and independently steered vehicle, where F_Limin≤F_Li≤F_{Li max}, F_Timin≤F_Ti≤F_{Ti max}, (i−1, . . . , 4), F_Limin and F_Timin respectively represent a minimum longitudinal force and a minimum lateral force of each wheel, and F_{Li max} and F_{Ti max} respectively represent a maximum longitudinal force and a maximum lateral force of each wheel. There is often a nonlinear or linear constraint relationship between the components in the u_v.
  • 3) All values of the u_v constitute a control set Ω_v={u_v} of the four-wheel independently driven and independently steered vehicle.
  • 4) A specific group of data u_vs ∈Ω_V in the Ω_V is processed by a wheel force control efficiency matrix B_V to generate a specific overall longitudinal force F_LS of the vehicle, a a specific overall lateral force F_TS of the vehicle. and a specific overall yawing moment M_S of specific overall lateral force the vehicle, which is denoted as V_VS=(F_LS, F_TS, M_S)^T. Therefore, v_VS=B_V·u_VS.
  • 5) All values of the v_VS constitute a control attainable set Φ_V of the four-wheel independently driven and independently steered vehicle, namely,

${\Phi }_V=\left\{v_{\mathrm{VS}}❘v_{\mathrm{VS}}=B_V·u_{\mathrm{VS}},u_{\mathrm{VS}}\in {\Omega }_V\right\}.$

The literature “Attainable Moments for the Constrained Control Allocation Problem” and the patent “Control Allocation Method for Overdrive System Based on Geometrically and Intuitively Constructed Attainable Set” (Patent Application No.: ZL201810131251.1) disclose a problem of determining a control attainable set when control actions of various actuators are independent of each other, in other words, when there is no constraint relationship between u_i and u_j(1≤i≤m, and 1≤j≤m, i≠j). The control attainable set is mathematically represented as follows:

$\begin{array}{cc}\Phi =\left\{v❘v=B·u,u={\left(u_1,\dots ,u_m\right)}^T,u\in \Omega \right\}s.t.u_{\mathrm{imin}}\le u_i\le u_{\mathrm{imax}},i=1,L,m& \left(1-2\right)\end{array}$

The patent “Method for Determining Control Attainable Set of Overdrive System under One Pair of Linear Constraint Control Components” (Patent Application No.: ZL201911405624.0) solves a problem of determining a control attainable set of a redundant drive system with a linear constraint relationship between only one pair of actuators, but is not applicable to determining a control attainable set of a redundant drive system with a linear constraint relationship between a plurality of pairs of actuators. The patent “Method for Determining Control Attainable Set of Overdrive System under Each Pair of Linear Constraint Control Components” (Patent Application No.: ZL201911411744.1) and the patent “Method for Determining Control Attainable Set of Overdrive System under a Plurality of Pairs of Linear Constraint Control Components (Patent Application No.: ZL201911405939.5) solve a problem of determining a control attainable set when there is a constraint relationship between each actuator pair and when there is a constraint relationship among a plurality of pairs of actuators. However, the two patents require determining a quasi-key boundary surface first. For each quasi-key boundary surface, a vertex mapped onto a control attainable set is first calculated to obtain a plane equation, and then a vertex of a boundary surface that has been determined as a key boundary surface is mapped to obtain a vertex of the control attainable set, and the vertex of the control attainable set is substituted into the plane equation. Finally, whether the quasi-key boundary surface is the key boundary surface is determined based on a characteristic that the control attainable set is a convex set. Calculation steps are complex and error-prone, and as a result, a good application result cannot be achieved.

SUMMARY

An objective of the present disclosure is to provide a method for calculating a control attainable set of a redundant drive system under a linear constraint to overcome the shortcomings in the existing technologies. The present disclosure solves a problem of determining a control attainable set of a parallel configuration redundant drive system in which the control attainable set is three-dimensional (3D) space, any three columns of a control efficiency matrix are linearly independent, and each pair of control components are linear constraint control components. The method can be used for evaluating a control capability of a parallel hybrid system with a redundant driving characteristic and each pair of control components being linear constraint control components in an advanced satellite, aircraft, ship, automobile, parallel robot, and the like. The present disclosure can provide a basis for control allocation of the system and be used for fault-tolerant control of the system after some actuators fail.

The present disclosure proposes a method for calculating a control attainable set of a redundant drive system under a linear constraint, including the following steps.

• • 1) establishing an expression of a control attainable set of a redundant drive system under each pair of linear constraint control components as follows:

$\begin{array}{cc}\Phi =\left\{v❘v=B·u,u={\left(u_1,\dots ,u_m\right)}^T,u\in \Omega \right\}s.t.u_{\mathrm{imin}}\le u_i\le u_{\mathrm{imax}},i=1,L,m\left(u_{\mathrm{kmax}}-u_{\mathrm{kmin}}\right)u_{k+1}+\left(u_{k+1\max }-u_{k+1\min }\right)u_k\le u_{k+1\max }{u}_{k\max }-u_{k+1\min }{u}_{k\min },k=1,3,\dots ,2\left[\frac{m}{2}\right]-1& \left(1\right)\end{array}$

• • where [m/2] represents an integer part obtained by dividing m by 2, u represents a control vector, the u is denoted as u_ELR, u_ELR=(u₁, . . . , u_m)^T, u_imin≤u_i≤u_{i max}, i=1,L,m (u_{k max}−u_{k min}))u_k+1+(u_k+1max−u_k+1min)u_k≤u_k+1maxu_{k max}−u_k+1minu_{k min}, and k=1, 3, . . . 2[m/2]−1;

where i^th control component u_i is a control action of a corresponding i^th actuator, where 1≤i≤m, and m represents a quantity of actuators; u_imin represents a minimum constraint value of the control action of the i^th actuator, and u_imax represents a maximum constraint value of the control action of the i^th actuator; Ω represents a control set, and Ω={u}; v represents a control attainable vector of the redundant drive system, and v=(v₁, v₂, v₃)^T, which represents a control output of the redundant drive system; Φ represents the control attainable set; and B represents a 3-row and m-column control efficiency matrix; and

• • the expression (1) is rewritten to a following expression (2):

$\begin{array}{cc}{\Phi }_{\mathrm{ELR}}=\left\{v❘v=B·u_{\mathrm{ELR}},u_{\mathrm{ELR}}={\left(u_1,\dots ,u_m\right)}^T,u_{\mathrm{ELR}}\in {\Omega }_{\mathrm{ELR}}\right\}s.t.u_{\mathrm{imin}}\le u_i\le u_{\mathrm{imax}},i=1,L,m;\left(u_{\mathrm{kmax}}-u_{\mathrm{kmin}}\right)u_{k+1}+\left(u_{k+1\max }-u_{k+1\min }\right)u_k\le u_{k+1\max }{u}_{\mathrm{kmax}}-u_{k+1\min }{u}_{\mathrm{kmin}},k=1,3,\dots ,2\left[\frac{m}{2}\right]-1& \left(2\right)\end{array}$

• • where u_k and u_k+1 represent a pair of control components with a linear constraint, Φ_ELR represents a control attainable set of a redundant drive system with each pair of control components being linear constraint control components, Ω_ELR represents a control set, u_ELR represents a control vector, and Ω_ELR={u_ELR};
  • 2) classifying all boundary surfaces of the control set Ω_ELR obtained in the step 1) into four types, which includes:
  • using, as a vertex of the control set Ω_ELR a point obtained by taking a corresponding maximum or minimum constraint value of each component u_i of the control vector u_ELR=(u₁, . . . , u_m)^T, where the m-dimensional control set Ω_ELR represented in the expression (2) has

$3^{\left[\frac{m}{2}\right]}·2^{\left(\frac{m}{2}\right)}$

vertices, and [m/2] represents a remainder of dividing m by 2; and

• • setting ∂(Ω_ELR) to represent a boundary of the Ω_ELR, and classifying boundary surfaces of the ∂(Ω_ELR) into a type-rectangular boundary surface, a type-rectangular boundary surface, a type-=2 \*ROMAN I rectangular boundary surface, and a triangular boundary surface based on a value of each component of each vertex, which specifically includes:
  • assuming that four vertices of the rectangular boundary surface are denoted as A, B, D, and C in clockwise order, and three vertices of the triangular boundary surface are denoted as A, C, and B in clockwise order,
  • if any rectangular boundary surface satisfies following conditions: for the vertices A and B, one component has a different value, and other components have a same value; for the vertices A and C, one component of another pair of components has a different value, and other components have a same value; and for the vertices A and D, the above two components have different values, and other components have a same value, determining that the rectangular boundary surface is the type-rectangular boundary surface;
  • if any rectangular boundary surface satisfies following conditions: for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; for the vertex C and the vertex D, the one pair of linear constraint components have different values, and other components have a same value; and for the vertex A and the vertex C, the one pair of linear constraint components have a same value, and other components have a same value except that one of the other components has a different value, determining that the rectangular boundary surface is the type-rectangular boundary surface;
  • if any rectangular boundary surface satisfies following conditions: for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; for the vertex A and the vertex C, another pair of linear constraint components have different values, and other components have a same value; and for the vertex D and the vertex A, each of the above two pairs of linear constraint components has a different value, and other components have a same value, determining that the rectangular boundary surface is the type-=2 \*ROMAN I rectangular boundary surface; and
  • if any right angle satisfies following conditions: for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value;
  • for the vertex A and the vertex C, one component of the one pair of linear constraint components has a different value, and other components have a same value; and for the vertex B and the vertex C, the other component of the one pair of linear constraint components has a different value, and other components have a same value, determining that the right angle is the triangular boundary surface;
  • 3) dividing all the boundary surfaces of the control set into groups, where
  • if values of two components in the u_ELR are between the corresponding minimum and maximum values, and values of other m−2 components are the corresponding minimum or maximum value, the m components form 2^m−2 boundary surfaces of the control set; and
  • if any two components in the u_ELR are denoted as a p^th component and a q^th component, values of the p^th component and the q^th component are between the corresponding minimum and maximum values, 1≤p≤m, 1≤q≤m, p<q values of other m−2 components are the corresponding minimum or maximum value, and a formed boundary surface is denoted as a P−q group, all the boundary surfaces of the control set are divided into the C_m² groups;
  • 4) determining a key boundary surface for each of the C_m² groups in the step 3), where
  • a boundary surface of an image mapped onto the Φ_ELR in the Ω_ELR of a boundary ∂(Φ_ELR) is denoted as the key boundary surface, the Ω is denoted as a set of key boundary surfaces, the Ω is initialized as an empty set, and the step includes following substeps:
  • 4-1) selecting any group for which no key boundary surface has been determined, denoting the group as a current p-q group, and performing a step 4-2);
  • 4-2) determining whether the p^th control component and the q^th control component are a pair of linear constraint components; and
  • if they are a pair of linear constraint components, performing a step 4-3);
  • if they are not a pair of linear constraint components, and m is an even number, performing a step 4-4);
  • if they are not a pair of linear constraint components, m is an odd number, and the p^th and q^th control components each have a paired linear constraint component, performing a step 4-4); or
  • if they are not a pair of linear constraint components, m is an odd number, and the q^th control component has no paired linear constraint component, performing a step 4-5);
  • 4-3) when the p^th control component and the q^th control component are a pair of linear constraint components, determining a key boundary surface of the p-q group according to a following specific method:
  • 4-3-1) when the p^th control component and the q^th control component are a pair of linear constraint components, determining that all boundary surfaces of the group are triangular boundary surfaces, where a value of each component of each point on the boundary surface satisfies a following formula:

$\begin{array}{cc}\{\begin{array}{c}{u}_{\mathrm{pmin}}\le u_p\le u_{\mathrm{pmax}}\\ u_{\mathrm{qmin}}\le u_q\le \frac{u_{\mathrm{qmax}}{u}_{\mathrm{pmax}}-u_{\mathrm{qmin}}{u}_{\mathrm{pmin}}-\left(u_{\mathrm{qmax}}-u_{\mathrm{qmin}}\right)u_p}{\left(u_{\mathrm{pmax}}-u_{\mathrm{pmin}}\right)}\\ u_i\in \left\{u_{\mathrm{imin}},u_{\mathrm{imax}}\right\},i\in Z^{+},1\le i\le m,i\ne p,q\end{array}& \left(3\right)\end{array}$

• • where a vertex of each boundary surface in the group is represented by an m-row and 3-column matrix Δ, and 3 columns are vectors corresponding to 3 vertices; values of the p^th and q^th components of three vertices on each boundary surface are shown in p^th and q^th rows of the matrix Δ, while values of other components are all the u_imax or the u_imin; one pair of linear constraint components in the other components do not simultaneously take a maximum constraint value corresponding to the one pair of linear constraint components; and the p^th row of the matrix Δ is (u_pmax, u_pmin, u_pmin), and the q^th row is (u_qmin, u_qmin, u_qmax);

$\Delta =\left[\begin{array}{ccc}\dots & \dots & \dots \\ u_{\mathrm{pmax}}& u_{\mathrm{pmin}}& u_{\mathrm{pmin}}\\ u_{\mathrm{qmin}}& u_{\mathrm{qmin}}& u_{\mathrm{qmax}}\\ \dots & \dots & \dots \end{array}\right];$

• • 4-3-2) constructing a coordinate rotation transformation ¹T, such that a transformed coordinate axis v₁ is perpendicular to an image of the triangular boundary surface of the group, which specifically includes: setting

${ }^{1}T=\left[\begin{array}{ccc}{ }^1{t}_{11}& { }^1{t}_{12}& { }^1{t}_{13}\\ { }^1{t}_{21}& { }^1{t}_{22}& { }^1{t}_{23}\\ { }^1{t}_{31}& { }^1{t}_{32}& { }^1{t}_{33}\end{array}\right],B=\left[\begin{array}{c}{b}_{11}\dots b_{1p}\dots b_{1q}\dots b_{1m}\\ b_{21}\dots b_{2p}\dots b_{2q}\dots b_{2m}\\ b_{31}\dots b_{3p}\dots b_{3q}\dots b_{3m}\end{array}\right],\mathrm{and}{ }^{1}C=\left[\begin{array}{c}{ }^1{c}_{11}\dots { }^1{c}_{1p}\dots { }^1{c}_{1q}\dots { }^1{c}_{1m}\\ { }^1{c}_{21}\dots { }^1{c}_{2p}\dots { }^1{c}_{2q}\dots { }^1{c}_{2m}\\ { }^1{c}_{31}\dots { }^1{c}_{3p}\dots { }^1{c}_{3q}\dots { }^1{c}_{3m}\end{array}\right],$

where ¹c_ij represents an element of an i^th row and a j^th column of the matrix ¹C, ¹t_ij represents an element of an i^th row and a j^th column of the matrix ¹T, and b_ij represents an element of an i^th row and a j^th column of the matrix B;

• • setting ¹C=¹T·B, such that ¹c_1p=0 and ¹c_1q=0;
  • substituting the ¹T and the B into the ¹C=¹T·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{ }^1{t}_{1s}·b_{\mathrm{sp}}=0\\ \sum _{s=1}^3{ }^1{t}_{1s}·b_{\mathrm{sq}}=0\end{array}& \left(4\right)\end{array}$

• • solving the linear equation set (4) to obtain a first row (¹t₁₁, ¹t₁₂, ¹t₁₃) of the ¹T; and
  • calculating a first row (¹c₁₁, . . . , ¹c_1m) of the matrix ¹C according to the ¹C=T·B;
  • 4-3-3) processing other rows than the rows p, q of the matrix Δ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and 1≤i≤m,i is an integer, i≠p, q; and
  • the rule is as follows:
  • when ¹c_1i·¹c_1i+1<0,
  • if ¹c_1i>0 and ¹c_1i+1<0, setting an i^th row of the matrix Δ to the u_imax and an i+1^th row to u_{i+1 min}; or
  • if ¹c_1i<0 and ¹c_1i+1<0, setting an i^th row of the matrix Δ to the u_imin and an i+1^th row to u_i+1max;
  • when ¹c_1i<0 and ¹c_1i+1<0,
  • setting an i^th row of the matrix Δ to the u_imin and an i+1^th row to u_i+1min; or
  • when ¹c_1i>0 and ¹c_1i+1<0,
  • calculating ¹z₀₁=¹c_1i·u_imax+¹c_1i+1·u_i+1min and ¹z₀₂=¹c_1i·u_imin+¹c_1i+1·u_i+1max, and then performing determining as follows:
  • if ¹z₀₁>¹z₀₂, setting an i^th row of the matrix Δ to the u_imax and an i+1^th row to u_i+1min; if ¹z₀₁<¹z₀₂ setting an i^th row of the matrix Δ to the u_imin and an i+1^th row to u_i+1max; or if ¹z₀₁=¹z₀₂, setting an i^th row and an i+1^th row of the matrix Δ as follows: setting the i^th row of the matrix Δ to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Δ to the u_imin and the i+1^th row to u_i+1max;
  • 4-3-4) processing other rows than the rows p, q of the matrix Δ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then performing a step 4-6), where 1≤i≤m,i is an integer, i≠p, q; and
  • the rule is as follows:
  • when ¹c_1i·¹c_1i+1<0,
  • if ¹c_1i>0 and ¹c_1i+1<0, setting the i^th row of the matrix Δ to the u_imin and the i+1^th row to the u_i+1max; or
  • if ¹c_1i<0 and ¹c_1i+1>0, setting the i^th row of the matrix Δ to the u_imax and the i+1^th row to u_{i+1 min};
  • when ¹c_1i>0 and ¹c_1i+1>0,
  • setting the i^th row of the matrix Δ to the u_imin and the i+1^th row to the u_i+1min; or
  • when ¹c_1i<0 and ¹c_1i+1<0,
  • calculating ²z₀₁=¹c_1i·u_imax+¹c_1i+1·u_i+1min and ²z₀₂=¹c_1i·u_imin+¹c_1i+1·u_i+1max and performing determining as follows: if ²z₀₁>²z₀₂, setting the i^th row of the matrix Δ to the u_imin and the i+1^th row to the u_imax; if ²z₀₁<²z₀₂, setting the i^th row of the matrix Δ to the u_imax and the i+1^th row to the u_i+1min; or if ¹z₀₁=¹z₀₂, setting the i^th row and the i+1^th row of the matrix Δ as follows: setting the i^th row of the matrix Δ to the u_imax and the i+1^th row to the u_i+1min; or setting the i^th row of the matrix Δ to the u_imin and the i+1^th row to the u_i+1max;
  • 4-4) when the p^th control component and the q^th control component are not a pair of linear constraint components, and each have the paired linear constraint component, determining a key boundary surface of the p-q group according to a following specific method:
  • 4-4-1) denoting the component paired with the p^th component as a p component, and the component paired with the q^th component as a q′^th component, where the p-q group has three types of rectangular boundary surfaces, and each component of each point on the boundary surfaces of the group has a value that satisfies a following formula:

$\begin{array}{cc}\{\begin{array}{c}{u}_{p\min }\le u_p\le u_{p\max }\\ u_{q\min }\le u_q\le u_{q\max }\\ u_{p^{\prime }}\in \left\{u_{p^{\prime }\min },\frac{u_{p^{\prime }\max }{u}_{p\max }-u_{p^{\prime }\min }{u}_{p\min }-\left(u_{p^{\prime }\max }-u_{p^{\prime }\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}\right\}\\ u_{q^{\prime }}\in \left\{u_{q^{\prime }\min },\frac{u_{q^{\prime }\max }{u}_{q\max }-u_{q^{\prime }\min }{u}_{q\min }-\left(u_{q^{\prime }\max }-u_{q^{\prime }\min }\right)u_q}{\left(u_{q\max }-u_{q\min }\right)}\right\}\\ u_i\in \left\{u_{i\min },u_{i\max }\right\},1\le i\le m,i\ne p,q,p^{\prime },q^{\prime }\\ u_k+u_{k+1}<u_{k\max }+u_{k+1\max },k=1,3,\dots ,2\left[\frac{m}{2}\right]-1,k\ne p,q,p^{\prime },q^{\prime }\end{array}& \left(5\right)\end{array}$

• • when u_p′=u_p′min and u_q′=u_q′min, determining that a boundary surface formed by a point satisfying the formula (5) is the type-I rectangular boundary surface;
  • when u_p′=u_p′min and

$u_{q^{\prime }}=\frac{u_{q^{\prime }\max }{u}_{q\max }-u_{q^{\prime }\min }{u}_{q\min }-\left(u_{q^{\prime }\max }-u_{q^{\prime }\min }\right)u_q}{\left(u_{q\max }-u_{q\min }\right)},$

determining that a boundary surface formed by a point satisfying the formula (5) is the type-II rectangular boundary surface;

• • when

$u_{p^{\prime }}=\frac{u_{p^{\prime }\max }{u}_{p\max }-u_{p^{\prime }\min }{u}_{p\min }-\left(u_{p^{\prime }\max }-u_{p^{\prime }\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}$

and u_q′=u_q′min, determining that a boundary surface formed by a point satisfying the formula (5) also is the type-II rectangular boundary surface;

• • when

$u_{p^{\prime }}=\frac{u_{p^{\prime }\max }{u}_{p\max }-u_{p^{\prime }\min }{u}_{p\min }-\left(u_{p^{\prime }\max }-u_{p^{\prime }\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}\mathrm{and}$ $u_{q^{\prime }}=\frac{u_{q^{\prime }\max }{u}_{q\max }-u_{q^{\prime }\min }{u}_{q\min }-\left(u_{q^{\prime }\max }-u_{q^{\prime }\min }\right)u_q}{\left(u_{q\max }-u_{q\min }\right)},$

determining that a boundary surface formed by a point satisfying the formula (5) is the type-III rectangular boundary surface; and

• • representing a vertex of each rectangular boundary surface of the group by using an m-row and 4-column matrix, where 4 columns respectively represent vectors corresponding to 4 vertices; a matrix Λ represents a vertex of one type-I rectangular boundary surface, a matrix ¹Π or ²Π represents a vertex of one type-II rectangular boundary surface, and a matrix T represents a vertex of one type-III rectangular boundary surface; values of the p^th, p′^th, q^th, and q′^th components of four vertices of each boundary surface are shown in a p^th row, a p′^th row, a q^th row, and a q′^th row of the matrix, and other components are all the u_imax or the u_imin; one pair of linear inequality constraint components do not simultaneously take a maximum constraint value corresponding to the one pair of linear inequality constraint components; for the Λ, the p^th row is (u_pmax, u_pmin, u_pmin, u_pmax), the p′^th row is (u_p′min, u_p′min, u_p′min, u_p′min), the q^th row is (u_qmin, u_qmin, u_qmax, u_qmax), and the q′^th row is (u_q′min, u_q′min, u_q′min, u_q′min);
  • for the ¹Λ, the p^th row is (u_pmax, u_pmin, u_pmin, u_pmax), the p′^th row is (u_p′min, u_p′max, u_p′max, u_p′min), the q^th row is (u_qmin, u_qmin, u_qmax, u_qmax), and the q′^th row is (u_q′min, u_q′min, u_q′min, u_q′min);
  • for the ²Π, the p^th row is (u_pmax, u_pmin, u_pmin, u_pmax), the p′^th row is (u_p′min, u_p′min, u_p′min, u_p′min, the q^th row is (u_qmin, u_qmin, u_qmax, u_qmax), and the q′^th row is (u_q′max, u_q′max, u_q′min, u_q′min);
  • for the T, the p^th row is (u_pmax, u_pmin, u_pmin, u_pmax), the p′^th row is (u_p′min, u_p′max, u_p′max, u_p′min), the q^th row is (u_qmin, u_qmin, u_qmax, u_qmax), and the q′^th row is (u_q′max, u_q′max, u_q′min, u_q′min);

$\begin{array}{cc}\Lambda =\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{\prime }\min }& u_{p^{\prime }\min }& u_{p^{\prime }\min }& u_{p^{\prime }\min }\\ \dots & \dots & \dots & \dots \\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{\prime }\min }& u_{q^{\prime }\min }& u_{q^{\prime }\min }& u_{q^{\prime }\min }\\ \dots & \dots & \dots & \dots \end{array}\right],& \left(6\right)\end{array}$ ${ }^1\Pi =\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{\prime }\min }& u_{p^{\prime }\max }& u_{p^{\prime }\max }& u_{p^{\prime }\min }\\ \dots & \dots & \dots & \dots \\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{\prime }\min }& u_{q^{\prime }\min }& u_{q^{\prime }\min }& u_{q^{\prime }\min }\\ \dots & \dots & \dots & \dots \end{array}\right],$ ${ }^2\Pi =\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{\prime }\min }& u_{p^{\prime }\min }& u_{p^{\prime }\min }& u_{p^{\prime }\min }\\ \dots & \dots & \dots & \dots \\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{\prime }\max }& u_{q^{\prime }\max }& u_{q^{\prime }\min }& u_{q^{\prime }\min }\\ \dots & \dots & \dots & \dots \end{array}\right],$ $T=\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{\prime }\min }& u_{p^{\prime }\max }& u_{p^{\prime }\max }& u_{p^{\prime }\min }\\ \dots & \dots & \dots & \dots \\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{\prime }\max }& u_{q^{\prime }\max }& u_{q^{\prime }\min }& u_{q^{\prime }\min }\\ \dots & \dots & \dots & \dots \end{array}\right],$

• • where if P is an odd number, the p^th row comes before the p′^th row; if p is an even number, the p^th row comes after the p′^th row; if q is an odd number, the q^th row comes before the q′^th row; and if q is an even number, the q^th row comes after the q′^th row;
  • 4-4-2) constructing a coordinate rotation transformation G, such that a transformed coordinate axis v₁ is perpendicular to an image of the type-III rectangular boundary surface, and images of all points on the boundary surface have equal coordinate values on the v₁ axis, which specifically includes:
  • setting

$G=\left[\begin{array}{ccc}{g}_{11}& g_{12}& g_{13}\\ g_{21}& g_{22}& g_{23}\\ g_{31}& g_{32}& g_{33}\end{array}\right],$ $B=\left[\begin{array}{ccccccccc}{b}_{11}& \dots & b_{1p}& b_{1p^{\prime }}& \dots & b_{1q}& b_{1q^{\prime }}& \dots & b_{1m}\\ b_{21}& \dots & b_{2p}& b_{2p^{\prime }}& \dots & b_{2q}& b_{2q^{\prime }}& \dots & b_{2m}\\ b_{31}& \dots & b_{3p}& b_{3p^{\prime }}& \dots & b_{3q}& b_{3q^{\prime }}& \dots & b_{3m}\end{array}\right],\mathrm{and}$ $E=\left[\begin{array}{ccccccccc}{e}_{11}& \dots & e_{1p}& e_{1p^{\prime }}& \dots & e_{1q}& e_{1q^{\prime }}& \dots & e_{1m}\\ e_{21}& \dots & e_{2p}& e_{2p^{\prime }}& \dots & e_{2q}& e_{2q^{\prime }}& \dots & e_{2m}\\ e_{31}& \dots & e_{3p}& e_{3p^{\prime }}& \dots & e_{3q}& e_{3q^{\prime }}& \dots & e_{3m}\end{array}\right],$

where g_ij represents an element of an i^th row and a j^th column of the matrix G, and e_ij represents an element of an i^th row and a j^th column of the matrix E;

• • setting E=G·B, such that

$e_{1p^{\prime }}=\frac{u_{p\max }-u_{p\min }}{u_{p^{\prime }\max }-u_{p^{\prime }\min }}{e}_{1p},e_{1q^{\prime }}=\frac{u_{q\max }-u_{q\min }}{u_{q^{\prime }\max }-u_{q^{\prime }\min }}{e}_{1q};$

and substituting the G and the B into the E=G·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{g}_{1s}·b_{\mathrm{sp}}=e_{1p}\\ \sum _{s=1}^3{g}_{1s}·b_{{\mathrm{sp}}^{\prime }}=\frac{u_{p\max }-u_{p\min }}{u_{p^{\prime }\max }-u_{p^{\prime }\min }}{e}_{1p}\\ \sum _{s=1}^3{g}_{1s}·b_{\mathrm{sq}}=e_{1q}\\ \sum _{s=1}^3{g}_{1s}·b_{{\mathrm{sq}}^{\prime }}=\frac{u_{q\max }-u_{q\min }}{u_{q^{\prime }\max }-u_{q^{\prime }\min }}{e}_{1q}\end{array}& \left(7\right)\end{array}$

• • solving the linear equation set (7) to obtain a first row of the G, and calculating a first row (e₁₁, . . . , e_1m) of the matrix E according to the E=G·B; and if e_1p′>0 and e_1q′>0, performing a step 4-4-3); if and e_1p′<0 and e_1q′<0, performing a step 4-4-4); or if e_1p′·e_1q′<0 performing a step 4-4-5);
  • 4-4-3) processing other rows than the rows p, q, p′, q′ of the matrix T according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then performing the step 4-4-5), where i=1, . . . , m, i≠p, p′, q, q′; and
  • the rule is as follows:
  • when e_1i·e_1i+1<0,
  • if e_1i>0 and e_1i−1<0, setting an i^th row of the matrix T to the u_imax and an i+1^th row to u_i+1min; or
  • if e_1i<0 and e_1i+1>0, setting an i^th row of the matrix T to the u_imin and an i+1^th row to u_i+1max;
  • when e_1i<0 and e_1i+1<0 ^{(i=1, . . . , m,i≠p, p′,q,q′)},
  • setting an i^th row of the matrix T to the u_imin and an i+1^th row to u_i+1min; or
  • when e_1i>0 and e_1i+1>0,
  • calculating ¹d₀₁=e_1i·u_imax+e_1i+1min and ¹d₀₂=e_1i·u_imin+e_1i+1max, and performing determining as follows: if ¹d₀₁>¹d₀₂, setting an i^th row of the matrix T to the u_imax and an i+1^th row to u_i+1min; if ¹d₀₁<¹d₀₂, setting an i^th row of the matrix T to the u_imin and an i+1^th row to u_i+1max; or if ¹d₀₁=¹d₀₂, setting an i^th row and an i+1^th row of the matrix T as follows: setting the i^th row of the matrix T to the u_imax and the i+1^th row to the u_i+1min; or setting the i^th row of the matrix T to the u_imin and the i+1^th row to the u_i+1max;
  • 4-4-4) processing other rows than the rows p, q, p′, q′ of the matrix T according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then performing the step 4-4-5), where i=1, . . . , m, i≠p, p′, q, q′; and
  • the rule is as follows:
  • when e_1i·e_1i+1<0,
  • if e_1i>0 and e_1i+1<0, setting an i^th row of the matrix T to the u_imin and an i+1^th row to u_i+1max; or
  • if e_1i<0 and e_1i+1>0, setting an i^th row of the matrix T to the u_imax and an i+1^th row to u_i+1min;
  • when e_1i<0 and e_1i+1<0,
  • calculating ²d₀₁=²h_1i·u_imax+²h_1i+1·+u_i+1min and ²d₀₂=²h_1i·u_imin+²h_1i+1·u_i+1max and if ²d₀₁>²d₀₂, setting an i^th row of the matrix T to the u_imin and an i+1^th row to u_i+1max; if ²d₀₁<²d₀₂, setting an i^th row of the matrix T to the u_imax and an i+1^th row to u_i+1min; or if ²d₀₁=²d₀₂, setting an i^th row and an i+1^th row of the matrix T as follows: setting the i^th row of the matrix T to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix T to the u_imin and the i+1^th row to u_i+1max; or
  • when e_1i>0 and e_1i+1>0,
  • setting an i^th row of the matrix T to the u_imin and an i+1^th row to u_i+1min;
  • 4-4-5) constructing a coordinate rotation transformation ²T, such that a transformed coordinate axis v₁ is perpendicular to an image of the type-I rectangular boundary surface of the group, which specifically includes: setting

${ }^{2}T=\left[\begin{array}{ccc}{ }_{}^2{t}_{11}^{}& { }_{}^2{t}_{12}^{}& { }_{}^2{t}_{13}^{}\\ { }_{}^2{t}_{21}^{}& { }_{}^2{t}_{22}^{}& { }_{}^2{t}_{23}^{}\\ { }_{}^2{t}_{31}^{}& { }_{}^2{t}_{32}^{}& { }_{}^2{t}_{33}^{}\end{array}\right],B=\left[\begin{array}{c}{b}_{11}\dots b_{1p}\dots b_{1q}\dots b_{1m}\\ b_{21}\dots b_{2p}\dots b_{2q}\dots b_{2m}\\ b_{31}\dots b_{3p}\dots b_{3q}\dots b_{3m}\end{array}\right],\mathrm{and}$ ${ }^{2}C=\left[\begin{array}{c}{ }_{}^2{c}_{11}^{}\dots { }_{}^2{c}_{1p}^{}\dots { }_{}^2{c}_{1q}^{}\dots { }_{}^2{c}_{1m}^{}\\ { }_{}^2{c}_{21}^{}\dots { }_{}^2{c}_{2p}^{}\dots { }_{}^2{c}_{2q}^{}\dots { }_{}^2{c}_{2m}^{}\\ { }_{}^2{c}_{31}^{}\dots { }_{}^2{c}_{3p}^{}\dots { }_{}^2{c}_{3q}^{}\dots { }_{}^2{c}_{3m}^{}\end{array}\right],$

where ²C_ij represents an element of an i^th row and a j^th column of the matrix ²C, and ²t_ij represents an element of an i^th row and a j^th column of the matrix ²T;

• • setting ²C=²T·B, such that ²c_1p=0 and ²C_1q=0; and substituting the ²T and the B into the ²C=²T·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{ }_{}^2{t}_{1s}·b_{\mathrm{sp}}=0\\ \sum _{s=1}^3{ }_{}^2{t}_{1s}·b_{\mathrm{sq}}=0\end{array}& \left(8\right)\end{array}$

• • solving the linear equation set (8) to obtain a first row (²t₁₁, ²t₁₂, ²t₁₃) of the ²T, and calculating a first row (²C₁₁, . . . , ²C_1m) of the matrix ²C according to the ²C=²T·B; and
  • if ²c_1p′<0 and ²C_1q′<0 performing a step 4-4-6); if ²c_1p′>0 and ²C_1q′>0, performing a step 4-4-7); or if ²c_1p′·²C_1q′<0, performing a step 4-4-8);
  • 4-4-6) processing other rows than the rows p, q, p′, q′ of the matrix Λ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then performing determining as follows:
  • if two key boundary surfaces have been found for the group, performing a step 4-6); otherwise, performing the step 4-4-8); where
  • the rule is as follows:
  • when ²c_1i·²c_1i+1<0,
  • if ²c_1i>0 and ²c_1i+1<0 setting an i^th row of the matrix Λ to the u_imax and an i+1^th row to u_i+1min; or
  • if ²c_1i<0 and ²c_1i+1>0, setting an i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1max;
  • when ²c_1i<0 and ²c_1i+1<0,
  • setting an i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1min; or
  • when ²c_1i>0 and ²c_1i+1>0,
  • calculating ³d₀₁=²c_1i·u_imax+²c_1i+1·u_i+1min, and ³d₀₂=²c_1i·u_imin+²c_1i+1·u_i+1max; and if ³d₀₁>³d₀₂, setting an i^th row of the matrix Λ to the u_imax and an i+1^th row to u_i+1min; if ³d₀₁<³d₀₂, setting an i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1max; or if ³d₀₁=³d₀₂, setting an i^th row and an i+1^th row of the matrix Λ as follows: setting the i^th row of the matrix Λ to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Λ to the u_imin and the i+1^th row to u_i+1max;
  • 4-4-7) processing other rows than the rows p, q, p′, q′ of the matrix according Λ to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then performing determining as follows:
  • if two key boundary surfaces have been found for the group, performing a step 4-6); otherwise, performing the step 4-4-8); where
  • the rule is as follows:
  • when ²c_1i·²c_1i+1<0,
  • if ²c_1i>0 and ²c_1i+1<0, setting an i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1max; or
  • if ²c_1i<0 and ²c_1i+1>0 setting an i^th row of the matrix Λ to the u_imax and an i+1^th row to u_i+1min;
  • when ²c_1i<0 and ²c_1i+1<0,
  • calculating ⁴d₀₁=²c_1i·u_imax+²c_1i+1·u_i+1min and ⁴d₀₂=²c_1i·u_imin+²c_1i+1·u_i+1max; and if ⁴d₀₁>⁴d₀₂, setting an i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1max; if ⁴d₀₁<⁴d₀₂, setting an i^th row of the matrix Λ to the u_imax and an i+1^th row to u_i+1min; or if ⁴d₀₁=⁴d₀₂, setting an i^th row and an i+1^th row of the matrix Λ as follows: setting the i^th row of the matrix Λ to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Λ to the u_imin and the i+1^th row to u_i+1max; or
  • when ²c_1i>0 and ²c_1i+1>0,
  • setting an i^th row of the matrix Λ to the u_imin and an i+1^th row to u_{i+1 min};
  • 4-4-8) constructing a coordinate rotation transformation ¹F, such that a transformed coordinate axis v₁ is Perpendicular to an Image of a Boundary Surface Whose Vertex Matrix is the ¹Π in the group, and images of all points on the boundary surface have equal coordinate values on the v₁ axis, which specifically includes: setting

${ }^{1}F=\left[\begin{array}{ccc}{ }_{}^1{f}_{11}^{}& { }_{}^1{f}_{12}^{}& { }_{}^1{f}_{13}^{}\\ { }_{}^1{f}_{21}^{}& { }_{}^1{f}_{22}^{}& { }_{}^1{f}_{23}^{}\\ { }_{}^1{f}_{31}^{}& { }_{}^1{f}_{32}^{}& { }_{}^1{f}_{33}^{}\end{array}\right],$ $B=\left[\begin{array}{cccccccc}{b}_{11}& \dots & b_{1p}& b_{1p^{\prime }}& \dots & b_{1q}& \dots & b_{1m}\\ b_{21}& \dots & b_{2p}& b_{2p^{\prime }}& \dots & b_{2q}& \dots & b_{2m}\\ b_{31}& \dots & b_{3p}& b_{3p^{\prime }}& \dots & b_{3q}& \dots & b_{3m}\end{array}\right],\mathrm{and}$ ${ }^{1}H=\left[\begin{array}{cccccccc}{ }_{}^1{h}_{11}^{}& \dots & { }_{}^1{h}_{1p}^{}& { }_{}^1{h}_{1p^{\prime }}^{}& \dots & { }_{}^1{h}_{1q}^{}& \dots & { }_{}^1{h}_{1m}^{}\\ { }_{}^1{h}_{21}^{}& \dots & { }_{}^1{h}_{2p}^{}& { }_{}^1{h}_{2p^{\prime }}^{}& \dots & { }_{}^1{h}_{2q}^{}& \dots & { }_{}^1{h}_{2m}^{}\\ { }_{}^1{h}_{31}^{}& \dots & { }_{}^1{h}_{3p}^{}& { }_{}^1{h}_{3p^{\prime }}^{}& \dots & { }_{}^1{h}_{3p}^{}& \dots & { }_{}^1{h}_{3m}^{}\end{array}\right],$

where ¹f_ij represents an element of an i^th row and a j^th column of the matrix ¹F, and ¹h_ij represents an element of an i^th row and a j^th column of the matrix ¹H;

• • setting ¹H=¹F·B, such that

${ }_{}^1{h}_{1p^{\prime }}^{}=\frac{u_{p\max }-u_{p\min }}{u_{p^{\prime }\max }-u_{p^{\prime }\min }}{ }_{}^1{h}_{1p}^{}$

and ¹_1q=0; and substituting the ¹F and the B into the ¹H= ¹F·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{ }_{}^1{f}_{1s}^{}·b_{\mathrm{sp}}={ }_{}^1{h}_{1p}^{}\\ \sum _{s=1}^3{ }_{}^1{f}_{1s}^{}·b_{{\mathrm{sp}}^{\prime }}=\frac{u_{p\max }-u_{p\min }}{u_{p^{\prime }\max }-u_{p^{\prime }\min }}{ }_{}^1{h}_{1p}^{}\\ \sum _{s=1}^3{ }_{}^1{f}_{1s}^{}·b_{\mathrm{sq}}=0\end{array}& \left(9\right)\end{array}$

• • solving the linear equation set (9) to obtain a first row of the ¹F, and calculating a first row (¹h₁₁, . . . , ¹h_1m) of the matrix ¹H according to the ¹H=¹F·B; and
  • if ¹h_1p′>0 ¹h_1q′<0, performing a step 4-4-9); if ¹h_1p′<0 and ¹h_1q′>0, performing a step 4-4-10); or if ¹h_1p′·¹h_1q′>0, performing a step 4-4-11);
  • 4-4-9) processing other rows than the rows p, q, P′, q′ of the matrix ¹Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then performing determining as follows:
  • if two key boundary surfaces have been found for the group, performing a step 4-6); otherwise, performing the step 4-4-11); where
  • the rule is as follows:
  • when ¹h_1i·¹h_1i+1<0,
  • if ¹h_1i>0 and ¹h_1i+1<0, setting an i^th row of the matrix ¹Π to the u_imax and an i+1^th row to u_i+1min; or
  • if ¹h_1i<0 and ¹h_1i+1>0, setting an i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1max;
  • when ¹h_1i<0 and ¹h_1i+1<0,
  • setting an i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1min; or
  • when ¹h_1i>0 and ¹h_1i+1>0,
  • calculating ⁵d₀₁=¹h_1i·u_imax+¹h_1i+1·u_i+1min and ⁵d₀₂=¹h_1i·u_imin+¹h_1i+1·u_i+1max; and if ⁵d₀₁>⁵d₀₂, setting an i^th row of the matrix ¹Π to the u_imax and an i+1^th row to u_i+1min; if ⁵d₀₁<⁵d₀₂, setting an i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1max; or if ⁵d₀₁=⁵d₀₂, setting an i^th row and an i+1^th row of the matrix ¹Π as follows: setting the i^th row of the matrix ¹Π to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix ¹Π to the u_imin and the i+1^th row to u_i+1max;
  • 4-4-10) processing other rows than the rows p, q, p′, q′ of the matrix ¹Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then performing determining as follows:
  • if two key boundary surfaces have been found for the group, performing a step 4-6); otherwise, performing the step 4-4-11); where
  • the rule is as follows:
  • when ¹h_1i·¹h_1i+1<0,
  • if ¹h_1i>0 and ¹h_1i+1<0, setting an i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1max; or
  • if ¹h_1i<0 and ¹h_1i+1>0, setting an i^th row of the matrix ¹Π to the u_imax and an i+1^th row to u_i+1min;
  • when ¹h_1i>0 and ¹h_1i+1>0,
  • setting an i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1min; or
  • when ¹h_1i<0 and ¹h_1i+1<0,
  • calculating ⁶d₀₁=¹h_1i·u_imax+¹h_1i+1·u_i+1min and ⁶d₀₂=¹h_1i·u_imin+¹h_1i+1·u_i+1max; and if ⁶d₀₁>⁶d₀₂, setting an i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1max; if ⁶d₀₁<⁶d₀₂, setting an i^th row of the matrix ¹Π to the u_imax and an i+1^th row to u_i+1min; or if ⁶d₀₁=⁶d₀₂, setting an i^th row and an i+1^th row of the matrix ¹Π as follows: setting the i^th row of the matrix ¹Π to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix ¹Π to the u_imin and the i+1^th row to u_i+1max;
  • 4-4-11) constructing a coordinate rotation transformation ²F, such that a transformed coordinate axis v₁ is perpendicular to an image of a boundary surface whose vertex matrix is the ²Π in the group, and images of all points on the boundary surface have equal coordinate values on the v₁ axis, which specifically includes:
  • setting

${ }^{2}F=\left[\begin{array}{ccc}{ }_{}^2{f}_{11}^{}& { }_{}^2{f}_{12}^{}& { }_{}^2{f}_{13}^{}\\ { }_{}^2{f}_{21}^{}& { }_{}^2{f}_{22}^{}& { }_{}^2{f}_{23}^{}\\ { }_{}^2{f}_{31}^{}& { }_{}^2{f}_{32}^{}& { }_{}^2{f}_{33}^{}\end{array}\right],$ $B=\left[\begin{array}{cccccccc}{b}_{11}& \dots & b_{1p}& b_{1p^{\prime }}& \dots & b_{1q}& \dots & b_{1m}\\ b_{21}& \dots & b_{2p}& b_{2p^{\prime }}& \dots & b_{2q}& \dots & b_{2m}\\ b_{31}& \dots & b_{3p}& b_{3p^{\prime }}& \dots & b_{3q}& \dots & b_{3m}\end{array}\right],\mathrm{and}$ ${ }^{2}H=\left[\begin{array}{cccccccc}{ }_{}^2{h}_{11}^{}& \dots & { }_{}^2{h}_{1p}^{}& \dots & { }_{}^2{h}_{1q}^{}& { }_{}^2{h}_{1q^{\prime }}^{}& \dots & { }_{}^2{h}_{1m}^{}\\ { }_{}^2{h}_{21}^{}& \dots & { }_{}^2{h}_{2p}^{}& \dots & { }_{}^2{h}_{2q}^{}& { }_{}^2{h}_{2q^{\prime }}^{}& \dots & { }_{}^2{h}_{2m}^{}\\ { }_{}^2{h}_{31}^{}& \dots & { }_{}^2{h}_{3p}^{}& \dots & { }_{}^2{h}_{3q}^{}& { }_{}^2{h}_{3q^{\prime }}^{}& \dots & { }_{}^2{h}_{3m}^{}\end{array}\right],$

where ²f_ij represents an element of an i^th row and a j^th column of the matrix ²F, and ²h_ij represents an element of an i^th row and a j^th column of the matrix ²H;

• • setting ²H=²F·B, such that

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{ }_{}^2{f}_{1s}^{}·b_{\mathrm{sp}}=0\\ \sum _{s=1}^3{ }_{}^2{f}_{1s}^{}·b_{\mathrm{sq}}={ }_{}^2{h}_{1q}^{}\\ \sum _{s=1}^3{ }_{}^2{f}_{1s}^{}·b_{{\mathrm{sq}}^{\prime }}=\frac{u_{q\max }-u_{q\min }}{u_{q^{\prime }\max }-u_{q^{\prime }\min }}{ }_{}^2{h}_{1q}^{}\end{array}& \left(10\right)\end{array}$

and ²h_1p=0 and substituting the ²F and the B into the ²H=²F·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{ }^2{f}_{1s}·b_{\mathrm{sp}}=0\\ \sum _{s=1}^3{ }^2{f}_{1s}·b_{\mathrm{sq}}={ }^2{h}_{1q} \\ \sum _{s=1}^3{ }^2{f}_{1s}·b_{{\mathrm{sq}}^{{ }_{}\prime }}=\frac{u_{q\max }-u_{q\min }}{u_{q^{{ }_{}\prime }\max }-u_{q^{{ }_{}\prime }\min }}{ }^2{h}_{1q}\end{array}& \left(10\right)\end{array}$

• • solving the linear equation set (10) to obtain a first row of the ²F, and calculating a first row (²h₁₁, . . . , ²h_1m) of the matrix ²H according to the ²H=²F·B; and if ²h_1p′<0 and ²h_1q′>0, performing a step 4-4-12); if ²h_1p′>0 and ²h_1q′<0 and, performing a step 4-4-13); or if ² h_1p′·²h_1q′>0, performing a step 4-6);
  • 4-4-12) processing other rows than the rows p, q, p′, q′ of the matrix ²Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then performing the step 4-6), where
  • the rule is as follows:
  • when ²h_1i·²h_1i+1<0,
  • if ²h_1i>0 and ²h_1i+1<0, setting an i^th row of the matrix ²Π to the u_imax and an i+1^th row to u_i+1min; or
  • if ²h_1i<0 and ²h_1i+1>0, setting an i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1max;
  • when ²h_1i<0 and ²h_1i+1<0,
  • setting an i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1min; or
  • when ²h_1i>0 and ²h_1i+1>0,
  • calculating ⁷d₀₁=²h_1i·u_imax+²h_1i+1·u_i+1min and ⁷d₀₂=²h_1i·u_imin+²h_1i+1·u_i+1max; and if ⁷d₀₁>⁷d₀₂, setting an i^th row of the matrix ²Π to the u_imax and an i+1^th row to u_i+1min; if ⁷d₀₁<⁷d₀₂, setting an i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1max; or if ⁷d₀₁=⁷d₀₂, setting an i^th row and an i+1^th row of the matrix ²Π as follows: setting the i^th row of the matrix ²Π to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix ²Π to the u_imin and the i+1^th row to u_i+1max;
  • 4-4-13) processing other rows than the rows p, q, p′, q′ of the matrix ²Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then performing the step 4-6), where
  • the rule is as follows:
  • when ²h_1i·²h_1i+1<0,
  • if ²h_1i>0 and ²h_1i+1<0, setting an i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1max; or
  • if ²h_1i<0 and ²h_1i+1>0, setting an i^th row of the matrix ²Π to the u_imax and an i+1^th row to u_i+1min;
  • when ²h_1i<0 and ²h_1i+1<0,
  • calculating ⁸d₀₁=²h_1i·u_imax+²h_1i+1·u_i+1min and ⁸d₀₂=²h_1i·u_imin+²h_1i+1·u_i+1max; and if ⁸d₀₁>⁸d₀₂, setting an i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1max; if ⁸d₀₁<⁸d₀₂, setting an i^th row of the matrix ²Π to the u_imax and an i+1^th row to u_i+1min; or if ⁸d₀₁=⁸d₀₂, setting an i^th row and an i+1^th row of the matrix ²Π as follows: setting the i^th row of the matrix ²Π to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix ²Π to the u_imin and the i+1^th row to u_i+1max; or
  • when ²h_1i>0 and ²h_1i+1>0,
  • setting an i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1min;
  • 4-5) when the p^th control component and the q^th control component are not a pair of linear constraint components, and the q^th control component has no paired linear constraint component, determining a key boundary surface of the p-q group according to a following specific method:
  • 4-5-1) denoting a component paired with the p^th component as a p′^th component, where the p-q group contains both the type-I and the type-II rectangular boundary surfaces, and each component of each point on the boundary surfaces of the group has a value that satisfies a following formula (11):

$\begin{array}{cc}\{\begin{array}{c}{u}_{\mathrm{pmin}}\le u_p\le u_{\mathrm{pmax}}\\ u_{\mathrm{qmin}}\le u_q\le u_{\mathrm{qmax}}\\ u_{p^{{ }_{}\prime }}\in \left\{u_{p^{{ }_{}\prime }\min },\frac{u_{p^{{ }_{}\prime }\max }{u}_{\mathrm{pmax}}-u_{p^{{ }_{}\prime }\min }{u}_{\mathrm{pmin}}-\left(u_{p^{{ }_{}\prime }\max }-u_{p^{{ }_{}\prime }\min }\right)u_p}{\left(u_{\mathrm{pmax}}-u_{\mathrm{pmin}}\right)}\right\}\\ u_i\in \left\{u_{\mathrm{imin}},u_{\mathrm{imax}}\right\},1\le i<m,i\ne p,q,p^{{ }_{}\prime }\\ u_k+u_{k+1}<u_{\mathrm{kmax}}+u_{k+1\max },k=1,3,\dots ,2\left[\frac{m}{2}\right]-1,k\ne p,q,p^{{ }_{}\prime }\end{array}& \left(11\right)\end{array}$

• • when u_p′=u_p′min determining that a boundary surface formed by a point satisfying the formula (11) is the type-I rectangular boundary surface; or when

$u_{p^{{ }_{}\prime }}=\frac{u_{p^{{ }_{}\prime }\max }{u}_{\mathrm{pmax}}-u_{p^{{ }_{}\prime }\min }{u}_{\mathrm{pmin}}-\left(u_{p^{{ }_{}\prime }\max }-u_{p^{{ }_{}\prime }\min }\right)u_p}{\left(u_{\mathrm{pmax}}-u_{\mathrm{pmin}}\right)},$

determining that a boundary surface formed by a point satisfying the formula (11) is the type-II rectangular boundary surface;

• • representing a vertex of each rectangular boundary surface of the group by using an m-row and 4-column matrix, where 4 columns respectively represent vectors corresponding to 4 vertices; a matrix Λ′ is used to represent a vertex of one type-I rectangular boundary surface, and a matrix Π′ represents a vertex of one type-II rectangular boundary surface; values of the p^th, p′^th, and q^th components of four vertices of each boundary surface are shown in a p^th row, a p′^th row, and a q^th row of the matrix, and other components are all u_imax or u_imin; one pair of linear inequality constraint components do not simultaneously take a maximum constraint value corresponding to the one pair of linear inequality constraint components; and for the Λ′, the p^th row is (u_pmax, u_pmin, u_pmin, u_pmax), the p′^th row is (u_p′min, u_p′min, u_p′min, u_p′min) the q^th row is (u_qmin, u_qmin, u_qmax, u_qmax), and the q′^th row is (u_q′min, u_q′min, u_q′min, u_q′min);

${\Lambda }^{\prime }=\left[\begin{array}{cccc}\cdots & \cdots & \cdots & \cdots \\ u_{\mathrm{pmax}}& u_{\mathrm{pmin}}& u_{\mathrm{pmin}}& u_{\mathrm{pmax}}\\ u_{p^{{ }_{}\prime }\min }& u_{p^{{ }_{}\prime }\min }& u_{p^{{ }_{}\prime }\min }& u_{p^{{ }_{}\prime }\min }\\ \cdots & \cdots & \cdots & \cdots \\ u_{\mathrm{qmin}}& u_{\mathrm{qmin}}& u_{\mathrm{qmax}}& u_{\mathrm{qmax}}\end{array}\right],$ ${\Pi }^{{ }_{}\prime }=\left[\begin{array}{cccc}\cdots & \cdots & \cdots & \cdots \\ u_{\mathrm{pmax}}& u_{\mathrm{pmin}}& u_{\mathrm{pmin}}& u_{\mathrm{pmax}}\\ u_{p^{{ }_{}\prime }\min }& u_{p^{{ }_{}\prime }\max }& u_{p^{{ }_{}\prime }\max }& u_{p^{{ }_{}\prime }\min }\\ \cdots & \cdots & \cdots & \cdots \\ u_{\mathrm{qmin}}& u_{\mathrm{qmin}}& u_{\mathrm{qmax}}& u_{\mathrm{qmax}}\end{array}\right]$

• • in the matrices shown in the above formula, if p is an odd number, the p^th row comes before the p′^th row; or if p is an even number, the p^th row comes after the p′^th row;
  • 4-5-2) constructing a coordinate rotation transformation T, such that a transformed coordinate axis v₁ is perpendicular to an image of the type-I rectangular boundary surface of the group, which specifically includes:
  • setting

${ }^{3}T=\left[\begin{array}{ccc}{ }^3{t}_{11}& { }^3{t}_{12}& { }^3{t}_{13}\\ { }^3{t}_{21}& { }^3{t}_{22}& { }^3{t}_{23}\\ { }^3{t}_{31}& { }^3{t}_{32}& { }^3{t}_{33}\end{array}\right],B=\left[\begin{array}{ccccc}{b}_{11}& \cdots & b_{1p}& \cdots & b_{1q}\\ b_{21}& \cdots & b_{2p}& \cdots & b_{2q}\\ b_{31}& \cdots & b_{3p}& \cdots & b_{3q}\end{array}\right],\mathrm{and}$ ${ }^{3}C=\left[\begin{array}{ccccc}{ }^3{c}_{11}& \cdots & { }^3{c}_{1p}& \cdots & { }^3{c}_{1q}\\ { }^3{c}_{21}& \cdots & { }^3{c}_{2p}& \cdots & { }^3{c}_{2q}\\ { }^3{c}_{31}& \cdots & { }^3{c}_{3p}& \cdots & { }^3{c}_{3q}\end{array}\right],$

where ³t_ij represents an element of an i^th row and a j^th column of the matrix ³T, and ³c_ij represents an element of an i^th row and a j^th column of the matrix ³C;

• • setting ³C=³T·B, such that ³c_1p=0 and ³c_1q=0; and substituting the ³T and the B into the ³C=³T·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{ }^3{t}_{1s}·b_{\mathrm{sp}}=0\\ \sum _{s=1}^3{ }^3{t}_{1s}·b_{\mathrm{sq}}=0 \end{array}& \left(12\right)\end{array}$

• • solving the linear equation set (12) to obtain a first row (³t₁₁, ³t₁₂, ³t₁₃) of the ³T, and calculating a first row (³c₁₁, . . . , ³c_1m) of the matrix ³C according to the ³C=3T·B; and if ³ c_1p′<0, performing a step 4-5-3); or if ³c_1p′>0, performing a step 4-5-4);
  • 4-5-3) processing other rows than the rows p, q, p′ of the matrix Λ′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then performing a step 4-5-5), where i=1, . . . , m, i≠p, p′, q; and
  • the rule is as follows:
  • when ³c_1i·³h_1i+1<0,
  • if ³c_1i>0 and ³c_1i+1<0, setting an i^th row of the matrix Λ′ to the u_imax and an i+1^th row to u_i+1min; or
  • if ³c_1i<0 and ³h_1i+1>0, setting an i^th row of the matrix Λ′ to the u_imin and an i+1^th row to u_i+1max;
  • when ³c_1i<0 and ³c_1i+1<0,
  • setting an i^th row of the matrix Π′ to the u_imin and an i+1^th row to u_i+1min; or
  • when ³c_1i>0 and ³h_1i+1>0,
  • calculating ⁹d₀₁=³c_1i·u_imax+³c_1i+1·u_i+1min, and ⁹d₀₂=³c_1i·u_imin+³c_1i+1·u_i+1max; and performing determining as follows:
  • if ⁹d₀₁>⁹d₀₂, setting an i^th row of the matrix Λ′ to the u_imax and an i+1^th row to u_i+1min; if ⁹d₀₁<⁹d₀₂, setting an i^th row of the matrix Λ′ to the u_imin and an i+1^th row to u_i+1max; or if ⁹d₀₁=⁹d₀₂, setting an i^th row and an i+1^th row of the matrix Λ′ as follows: setting the i^th row of the matrix Λ′ to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Λ′ to the u_imin and the i+1^th row to u_i+1max;
  • 4-5-4) processing other rows than the rows p, q, p′ of the matrix Λ′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then performing a step 4-5-5), where i=1, . . . , m, i≠p, p′, q; and
  • the rule is as follows:
  • when ³c_1i·³h_1i+1<0,
  • if ³c_1i>0 and ³c_1i+1<0, setting an i^th row of the matrix Λ′ to the u_imin and an i+1^th row to u_i+1max; or
  • if ³c_1i<0 and ³h_1i+1>0, setting an i^th row of the matrix Λ′ to the u_imax and an i+1^th row to u_i+1min;
  • when ³c_1i>0 and ³c_1i+1>0,
  • setting an i^th row of the matrix Λ′ to the u_imin and an i+1^th row to u_i+1min; or
  • when ³c_1i<0 and ³h_1i+1<0,
  • calculating ¹⁰d₀₁=³c_1i·u_imax+³c_1i+1·u_i+1min, and ¹⁰d₀₂=³c_1i·u_imin+³c_1i+1·u_i+1max; and if ¹⁰d₀₁>¹⁰d₀₂, setting an i^th row of the matrix Λ′ to the u_imin and an i+1^th row to u_i+1max; if ¹⁰d₀₁<¹⁰d₀₂, setting an i^th row of the matrix Λ′ to the u_imax and an i+1^th row to u_i+1min; or if ¹⁰d₀₁=¹⁰d₀₂, setting an i^th row and an i+1^th row of the matrix Λ′ as follows: setting the i^th row of the matrix Λ′ to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Λ′ to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Λ′ to the u_imin and the i+1^th row to u_i+1max;
  • 4-5-5) constructing a coordinate rotation transformation ³F, such that a transformed coordinate axis v₁ is perpendicular to an image of a boundary surface whose vertex matrix is the Π′ in the group, and images of all points on the boundary surface have equal coordinate values on the v₁ axis, which specifically includes:
  • setting

${ }^{3}F=\left[\begin{array}{ccc}{ }^3{f}_{11}& { }^3{f}_{12}& { }^3{f}_{13}\\ { }^3{f}_{21}& { }^3{f}_{22}& { }^3{f}_{23}\\ { }^3{f}_{31}& { }^3{f}_{32}& { }^3{f}_{33}\end{array}\right],B=\left[\begin{array}{cccccc}{b}_{11}& \cdots & b_{1p}& b_{1p^{{ }_{}\prime }}& \cdots & b_{1q}\\ b_{21}& \cdots & b_{2p}& b_{2p^{{ }_{}\prime }}& \cdots & b_{2q}\\ b_{31}& \cdots & b_{3p}& b_{3p^{{ }_{}\prime }}& \cdots & b_{3q}\end{array}\right],\mathrm{and}$ ${ }^{3}H=\left[\begin{array}{cccccc}{ }^3{h}_{11}& \cdots & { }^3{h}_{1p}& { }^3{h}_{1p^{{ }_{}\prime }}& \cdots & { }^3{h}_{1q}\\ { }^3{h}_{21}& \cdots & { }^3{h}_{2p}& { }^3{h}_{2p^{{ }_{}\prime }}& \cdots & { }^3{h}_{2q}\\ { }^3{h}_{31}& \cdots & { }^3{h}_{3p}& { }^3{h}_{3p^{{ }_{}\prime }}& \cdots & { }^3{h}_{3q}\end{array}\right],$

where ³h_ij represents an element of an i^th row and a j^th column of the matrix ³H, and ³f_ij represents an element of an i^th row and a j^th column of the matrix ³F;

• • setting ³H=³F·B, such that

${ }^3{h}_{1p^{{ }_{}\prime }}=\frac{u_{\mathrm{pmax}}-u_{\mathrm{pmin}}}{u_{p^{{ }_{}\prime }\max }-u_{p^{{ }_{}\prime }\min }}{ }^3{h}_{1p},$

and ³h_1q=0; and substituting the ³F and the B into the ³H=³F·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{ }^3{f}_{1s}·b_{\mathrm{sp}}={ }^3{h}_{1p}\\ \sum _{s=1}^3{ }^3{f}_{1s}·b_{{\mathrm{sp}}^{{ }_{}\prime }}=\frac{u_{p\max }-u_{p\min }}{u_{p^{{ }_{}\prime }\max }-u_{p^{{ }_{}\prime }\min }}{ }^3{h}_{1p} \\ \sum _{s=1}^3{ }^3{f}_{1s}·b_{\mathrm{sq}}=0\end{array}& \left(13\right)\end{array}$

• • solving the linear equation set (13) to obtain a first row of the ³F, and calculating a first row (³h₁₁, . . . , ³h_1m) of the matrix ³H according to the ³H=³F·B; and if ³h_1p′>0, performing a step 4-5-6); or if ³h_1p′<0, performing a step 4-5-7);
  • 4-5-6) processing other rows than the rows p, q, p′ of the matrix Π′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then performing a step 4-6), where i=1, . . . , m, i≠p, p′, q; and
  • the rule is as follows:
  • when ³h_1i·³h_1i+1<0
  • if ³h_1i>0 and ³h_1i+1<0, setting an i^th row of the matrix Π′ to the u_imax and an i+1^th row to u_i+1min; or
  • if ³h_1i<0 and ³h_1i+1>0, setting an i^th row of the matrix Π′ to the u_imin and an i+1^th row to u_i+1max;
  • when ³h_1i<0 and ³h_1i+1<0,
  • setting an i^th row of the matrix Π′ to the u_imin and an i+1^th row to u_i+1min; or
  • when ³h_1i; >0 and ³h_1i+1>0,
  • calculating ¹¹d₀₁=³c_1i·u_imax+³c_1i+1·u_i+1min, and ¹¹d₀₂=³c_1i·u_imin+³c_1i+1·u_i+1max; and if ¹¹d₀₁>¹¹d₀₂, setting an i^th row of the matrix Π′ to the u_imax and an i+1^th row to u_i+1min; if ¹¹d₀₁<¹¹d₀₂, setting an i^th row of the matrix Π′ to the u_imin and an i+1^th row to u_i+1max; or if ¹¹d₀₁=¹¹d₀₂, setting an i^th row and an i+1^th row of the matrix Π′ as follows: setting the i^th row of the matrix Π′ to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Π′ to the u_imin and the i+1^th row to u_i+1max;
  • 4-5-7) processing other rows than the rows p, q, p′ of the matrix Π′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then performing a step 4-6), where i=1, . . . , m, i≠p, p′, q; and
  • the rule is as follows:
  • when ³h_1i·³h_1i+1<0
  • if ³h_1i>0 and ³h_1i+1<0, setting an i^th row of the matrix Π′ to the u_imin and an i+1^th row to u_i+1max; or
  • if ³h_1i<0 and ³h_1i+1>0, setting an i^th row of the matrix Π′ to the u_imax and an i+1^th row to u_i+1min;
  • when ³h_1i<0 and ³h_1i+1<0,
  • calculating ¹²d₀₁=³c_1i·u_imax+³c_1i+1·u_i+1min, and ¹²d₀₂=³c_1i·u_imin+³c_1i+1·u_i+1max; and if ¹²d₀₁>¹²d₀₂, setting an i^th row of the matrix Π′ to the u_imin and an i+1^th row to u_i+1max; if ¹²d₀₁<¹²d₀₂, setting an i^th row of the matrix Π′ to the u_imax and an i+1^th row to u_i+1min; or if ¹²d₀₁=¹²d₀₂, setting an i^th row and an i+1^th row of the matrix Π′ as follows: setting the i^th row of the matrix Π′ to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Π′ to the u_imin and the i+1^th row to u_i+1max; or
  • when ³h_1i>0 and ³h_1i+1>0,
  • setting an i^th row of the matrix Π′ to the u_imin and an i+1^th row to u_i+1min; and
  • 4-6) returning to the step 4-1), selecting a next group for which no key boundary surface has been determined, until the key boundary surface has been determined for all groups, and then performing a step 5);
  • 5) checking all key boundary surfaces in the 52, and performing determining as follows: if there are identical key boundary surfaces, retaining only one of the identical key boundary surfaces, and removing other key boundary surfaces that are the same as the retained one key boundary surface; after processing all the key boundary surfaces, forming a new set of key boundary surfaces, which is denoted as Ω′; and
  • then performing a step 6); and
  • 6) mapping all vertices of each key boundary surface in the Ω′ according to v=B·u_ELR to obtain all vertices of a boundary surface corresponding to the control attainable set, so as to determine the boundary surface of the control attainable set, where all key boundary surfaces in the Ω′ are all the key boundary surfaces of the control set; the boundary surface of the control attainable set is a quadrilateral or triangle; and boundary surfaces that are of the control attainable set and determined based on all the key boundary surfaces in the Ω′ constitute the boundary ∂(Φ_ELR) of the control attainable set.

The present disclosure has following characteristics and beneficial effects:

The present disclosure breaks through a limitation that all control components need to be independent of each other in existing technologies, and solves a problem of determining a control attainable set of a parallel configuration redundant drive system in which the control attainable set is 3D space, any three columns of a control efficiency matrix are linearly independent, and each pair of control components are linear constraint control components. Instead, the present disclosure can omit a step of determining a quasi-key boundary surface but directly find a key boundary surface for each group. The method is concise and computationally efficient, and has high application value.

The method can be used for evaluating a control capability of a parallel hybrid system with a redundant driving characteristic and each pair of control components being linear constraint control components in an advanced satellite, aircraft, ship, automobile, parallel robot, and the like. The present disclosure can provide a basis for control allocation of the system and be used for fault-tolerant control of the system after some actuators fail.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an overall flowchart of a method for calculating a control attainable set of a redundant drive system under a linear constraint according to the present disclosure.

DETAILED DESCRIPTION OF THE EMBODIMENTS

The present disclosure provides a method for calculating a control attainable set of a redundant drive system under a linear constraint. The following provides further detailed description based on the accompanying drawings and specific embodiments.

The present disclosure proposes a method for calculating a control attainable set of a redundant drive system under a linear constraint. An overall process is shown in FIG. 1, and the method includes following steps.

1) Establish an expression of a control attainable set of a redundant drive system under each pair of linear constraint control components as follows:

$\begin{array}{cc}\Phi =\left\{v❘v=B·u,u={\left(u_1,\dots ,u_m\right)}^T,u\in \Omega \right\}s.t.& \left(1\right)\end{array}$ $u_{\mathrm{imin}}\le u_i\le u_{\mathrm{imax}},i=1,L,m$ $\left(u_{\mathrm{kmax}}-u_{\mathrm{kmin}}\right)u_{k+1}+\left(u_{k+1\max }-u_{k+1\min }\right)u_k\le u_{k+1\max }{u}_{\mathrm{kmax}}-u_{k+1\min }{u}_{\mathrm{kmin}},$ $k=1,3,\dots ,2\left[\frac{m}{2}\right]-1$

In the above expression, [m/2] represents an integer part obtained by dividing m by 2, u represents a control vector, the u is denoted as u_ELR, u_ELR=(u₁, . . . , u_m)^T, u_imin≤u_i≤u_imax, i=1,L,m, (u_kmax−u_kmin)u_k+1+(u_k+1max−u_k+1min)u_k≤u_k+1maxu_kmax−u_k+1minu_kmin, and k=1,3, . . . , 2[m/2]−1.

An i^th control component u_i is a control action of a corresponding i^th actuator, where 1≤i≤m, and m represents a quantity of actuators; and u_imin represents a minimum constraint value of the control action of the i^th actuator, and u_imax represents a maximum constraint value of the control action of the i^th actuator. Ω represents a control set, and Ω={u}; v represents a control attainable vector of the redundant drive system, and v=(v₁, v₂, v₃)^T, which represents a control output of the redundant drive system; Φ represents the control attainable set; and B represents a 3-row and m-column control efficiency matrix.

For convenience of subsequent description, the expression (1) is rewritten to an expression (2):

$\begin{array}{cc}{\Phi }_{\mathrm{ELR}}=\left\{v❘v=B·u_{\mathrm{ELR}},u_{\mathrm{ELR}}={\left(u_1,\dots ,u_m\right)}^T,u_{\mathrm{ELR}}\in {\Omega }_{\mathrm{ELR}}\right\}s.t.& \left(2\right)\end{array}$ $u_{\mathrm{imin}}\le u_i\le u_{\mathrm{imax}},i=1,L,m;$ $\left(u_{\mathrm{kmax}}-u_{\mathrm{kmin}}\right)u_{k+1}+\left(u_{k+1\max }-u_{k+1\min }\right)u_k\le u_{k+1\max }{u}_{\mathrm{kmax}}-u_{k+1\min }{u}_{\mathrm{kmin}},$ $k=1,3,\dots ,2\left[\frac{m}{2}\right]-1$

In the above expression, u_k and u_k+1 represent a pair of control components with a linear constraint, which is referred to as a pair of linear constraint control components, Φ_ELR represents a control attainable set of a redundant drive system with each pair of control components being linear constraint control components, Ω_ELR represents a control set, u_ELR represents a control vector, and Ω_ELR={u_ELR}. A boundary of the Ω_ELR is represented by ∂(Ω_ELR), and a boundary of the Φ_ELR is represented by ∂(Φ_ELR). In this way, determining of the control attainable set of the redundant drive system with each pair of control components being linear constraint control components is to determine ∂(Φ_ELR) based on the given Ω_ELR and B.

• • 2) Classify all boundary surfaces of the control set Ω_ELR obtained in the step 1) into four types.

In this embodiment, for the control set Ω_ELR={u_ELR}, a point obtained by taking a corresponding maximum or minimum constraint value of each component u_i(i=1, . . . , m) of the control vector u_ELR=(u₁, . . . , u_m)^T is used as a vertex of the control set Ω_ELR. m represents the quantity of actuators, and m>3. One m-dimensional control set has at most 2^m vertices. Because paired linear constraint control components cannot take the maximum value simultaneously, the m-dimensional control set Ω_ELR represented by the expression (2) has

$3^{\left[\frac{m}{2}\right]}·2^{\left(\frac{m}{2}\right)}$

vertices. [m/2] represents the integer part obtained by dividing m by 2, and (m/2) represents a remainder of dividing m by 2.

Ω_ELR represents the control set of the redundant drive system with each pair of control components being linear constraint control components, and the boundary of the Ω_ELR is the ∂(Ω_ELR) Based on a value of each component of each vertex, there are four types of boundary surfaces constituting the ∂(Ω_ELR): a type-rectangular boundary surface, a type-rectangular boundary surface, a type-=2\*ROMAN I rectangular boundary surface, and a triangular boundary surface. The four types of boundary surfaces are defined as follows:

Four vertices of the rectangular boundary surface are denoted as A, B, D, and C in clockwise order, and three vertices of the triangular boundary surface are denoted as A, C, and B in clockwise order.

A rectangle that satisfies following three conditions is referred to as the type-I rectangular boundary surface: (i) for the vertices A and B, one component has a different value, and other components have a same value; (ii) for the vertices A and C, one component of another pair of components has a different value, and other components have a same value; and (iii) for the vertices A and D, the above two components have different values, and other components have a same value.

A rectangle that satisfies following three conditions is referred to as the type-II rectangular boundary surface: (i) for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; (ii) for the vertex C and the vertex D, the one pair of linear constraint components have different values, and other components have a same value; and (iii) for the vertex A and the vertex C, the one pair of linear constraint components have a same value, and other components have a same value except that one of the other components has a different value.

A rectangle that satisfies following three conditions is referred to as the type−=2\*ROMAN I rectangular boundary surface: (i) for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; (ii) for the vertex A and the vertex C, another pair of linear constraint components have different values, and other components have a same value; and (iii) for the vertex D and the vertex A, each of the above two pairs of linear constraint components has a different value, and other components have a same value.

A right triangle that following three conditions is referred to as the triangular boundary surface: (i) for the vertex A and the vertex B, one pair of linear constraint components have different values, and other components have a same value; (ii) for the vertex A and the vertex C, one component of the one pair of linear constraint components have a different value, and other components have a same value; and (iii) for the vertex B and the vertex C, the other component of the one pair of linear constraint components has a different value, and other components have a same value.

3) Divide all the boundary surfaces of the control set Ω_ELR into C_m² groups.

If values of two components of the u_ELR are between the corresponding minimum and maximum values, and values of other m−2 components are the corresponding minimum or maximum value, the m components form 2^m−2 boundary surfaces of the control set.

If any two components in the u_ELR are denoted as a p^th component and a q^th component, values of the p^th component and the q^th component are between the corresponding minimum and maximum values, 1≤p≤m, 1≤q≤m, p<q, values of other m−2 components are the corresponding minimum or maximum value, and a boundary surface formed in such a manner is added to one group that is referred to as a p-q group, all the boundary surfaces in the control set are divided into the C_m² groups.

4) Determine a key boundary surface for each of the C_m² groups in the step 3).

A boundary surface of an image mapped onto the Φ_ELR in the Ω_ELR of the boundary ∂(Φ_ELR) is denoted as the key boundary surface, the Ω is denoted as a set of key boundary surfaces, and the Ω is initialized as an empty set.

4-1) Select any group for which no key boundary surface has been determined, denote the group as a current p-q group, and perform a step 4-2).

4-2) Determine whether the p^th control component and the q^th control component are a pair of linear constraint components; and

• • if they are a pair of linear constraint components, perform a step 4-3);
  • if they are not a pair of linear constraint components, and m is an even number, perform a step 4-4);

if they are not a pair of linear constraint components, m is an odd number, and the p^th and q^th control components each have a paired linear constraint component, perform a step 4-4); or

if they are not a pair of linear constraint components, m is an odd number, and the q^th control component has no paired linear constraint component, perform a step 4-5).

It should be noted that an m^th component does not have a paired linear constraint component only when m is the odd number. Since it has been limited earlier that p<q the p^th control component must have a paired linear constraint component.

4-3) When the p^th control component and the q^th control component are a pair of linear constraint components, determine a key boundary surface of the p-q group according to a following specific method:

• • 4-3-1) When the p^th control component and the q^th control component are a pair of linear constraint components, determine that all boundary surfaces of the group are triangular boundary surfaces, where a value of each component of each point on the boundary surface satisfies a following formula:

$\begin{array}{cc}\{\begin{array}{c}{u}_{\mathrm{pmin}}\le u_p\le u_{\mathrm{pmax}}\\ u_{\mathrm{qmin}}\le u_q\le \frac{u_{\mathrm{qmax}}{u}_{\mathrm{pmax}}-u_{\mathrm{qmin}}{u}_{\mathrm{pmin}}-\left(u_{\mathrm{qmax}}-u_{\mathrm{qmin}}\right)u_p}{\left(u_{\mathrm{pmax}}-u_{\mathrm{pmin}}\right)}\\ u_i\in \left\{u_{\mathrm{imin}},u_{\mathrm{imax}}\right\},i\in Z^{{ }_{}+},1\le i\le m,i\ne p,q\end{array}& \left(3\right)\end{array}$

A vertex of each boundary surface in the group is represented by an m-row and 3-column matrix 4, and ³ columns are vectors corresponding to 3 vertices. Values of the p^th and q^th components of three vertices on each boundary surface are shown in p^th and q^th rows of the matrix Δ. A value of an i(i=1, . . . , m,i≠p,q)^th component is the u_imax or the u_imin, and an i(i=1, . . . , m, i≠p, q)^th row of the corresponding matrix Δ is the u_imax Or the u_imin. However, one pair of linear constraint components in other components cannot simultaneously take a maximum constraint value corresponding to the one pair of linear constraint components, and values of the other components are uniformly represented by ellipses in the Δ. The p^th row of the matrix Δ is (u_pmax, u_pmin, u_pmin), and the q^th row is (u_qmin, u_qmin, u_qmax).

$\Delta =\left[\begin{array}{ccc}\cdots & \cdots & \cdots \\ u_{\mathrm{pmax}}& u_{\mathrm{pmin}}& u_{\mathrm{pmin}}\\ u_{\mathrm{qmin}}& u_{\mathrm{qmin}}& u_{\mathrm{qmax}}\\ \cdots & \cdots & \cdots \end{array}\right].$

4-3-2) Construct a coordinate rotation transformation ¹T, such that a transformed coordinate axis v₁ is perpendicular to an image of the triangular boundary surface of the group, where only a first row of the ¹T is required because only the V₁ axis is considered.

${ }^{1}T=\left[\begin{array}{ccc}{ }^1{t}_{11}& { }^1{t}_{12}& { }^1{t}_{13}\\ { }^1{t}_{21}& { }^1{t}_{22}& { }^1{t}_{23}\\ { }^1{t}_{31}& { }^1{t}_{32}& { }^1{t}_{33}\end{array}\right],B=\left[\begin{array}{ccccccc}{b}_{11}& \cdots & b_{1p}& \cdots & b_{1q}& \cdots & b_{1m}\\ b_{21}& \cdots & b_{2p}& \cdots & b_{2q}& \cdots & b_{2m}\\ b_{31}& \cdots & b_{3p}& \cdots & b_{3q}& \cdots & b_{3m}\end{array}\right],\mathrm{and}$ ${ }^{1}C=\left[\begin{array}{ccccccc}{ }^1{c}_{11}& \cdots & { }^1{c}_{1p}& \cdots & { }^1{c}_{1q}& \cdots & { }^1{c}_{1m}\\ { }^1{c}_{21}& \cdots & { }^1{c}_{2p}& \cdots & { }^1{c}_{2q}& \cdots & { }^1{c}_{2m}\\ { }^1{c}_{31}& \cdots & { }^1{c}_{3p}& \cdots & { }^1{c}_{3q}& \cdots & { }^1{c}_{3m}\end{array}\right]$

are set, where ¹c_ij represents an element of an i^th row and a j^th column of the matrix ¹C, ¹t_ij represents an element of an i^th row and a j^th column of the matrix ¹T, and b_ij represents an element of an i^th row and a j^th column of the matrix B.

¹C=¹T·B is set, such that ¹c_1p=0, and ¹c_1q=0. The ¹T and the B are substituted into the ¹C=¹T·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{ }^1{t}_{1s}·b_{sp}=0\\ \sum _{s=1}^3{ }^1{t}_{1s}·b_{sq}=0\end{array}& \left(4\right)\end{array}$

The linear equation set (4) is solved to obtain the first row (¹t₁₁, ¹t₁₂, ¹t₁₃) of the ¹T. Then, a first row (¹C₁₁, . . . , ¹C_1m) of the matrix ¹C is calculated according to the ¹C=¹T·B.

4-3-3) Process other rows than the rows P, q of the matrix Δ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and 1≤i≤m,i is an integer, i≠p,q.

The rule is as follows:

• • when ¹c_1i·¹c_1i+1<0,
  • if ¹c_1i>0 and ¹c_1i+1<0, setting the i^th row of the matrix Δ to the u_imax and an i+1^th row to u_i+1min; or
  • if ¹c_1i<0 and ¹c_1i+1>0, setting the i^th row of the matrix Δ to the u_imin and an i+1^th row to u_i+1max; or

when ¹c_1i<0 and ¹c_1i+1<0,

• • setting the i^th row of the matrix Δ to the u_imin and an i+1^th row to u_i+1min; or
  • when ¹c_1i>0 and ¹c_1i+1>0,
  • calculating ¹z₀₁=¹c_1i·u_imax+¹c_1i+1·u_i+1min and ¹z₀₂=¹c_1i·u_imin+¹c_1i+1·u_i+1max and then performing determining as follows:
  • if ¹z₀₁>¹z₀₂, setting an i^th row of the matrix Δ to the u_imax and an i+1^th row to u_i+1min; if ¹z₀₁<¹z₀₂, setting an i^th row of the matrix Δ to the u_imin and an i+1^th row to u_i+1max; or if ¹z₀₁=¹z₀₂, setting an i^th row and an i+1^th row of the matrix Δ as follows: setting the i^th row of the matrix Δ to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Δ to the u_imin and the i+1^th row to u_i+1max.
  • 4-3-4) Process other rows than the rows p, q of the matrix Δ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then perform a step 4-6), where 1≤i≤m, i is an integer, i≠p, q.

The rule is as follows:

• • when ¹c_1i·¹c_1i+1<0,
  • if ¹c_1i>0 and ¹c_1i+1<0, setting the i^th row of the matrix Δ to the u_imin and an i+1^th row to u_i+1max; or
  • if ¹c_1i<0 and ¹c_1i+1>0, setting the i^th row of the matrix Δ to the u_imax and an i+1^th row to u_i+1min; or
  • when ¹c_1i>0 and ¹c_1i+1>0,
  • setting the i^th row of the matrix Δ to the u_imin and an i+1^th row to u_i+1min; or
  • when ¹c_1i<0 and ¹c_1i+1<0,
  • calculating ²z₀₁=¹c_1i·u_imax+¹c_1i+1·u_i+1min and ²z₀₂=¹c_1i·u_imin+¹c_1i+1·u_i+1max and performing determining as follows: if ²z₀₁>²z₀₂, setting an i^th row of the matrix Δ to the u_imin and the i+1^th row to the u_i+1max; if ²z₀₁<²z₀₂, setting an i^th row of the matrix Δ to the u_imax and the i+1^th row to u_i+1min; or if ¹z₀₁=¹z₀₂, setting an i^th row and an i+1^th row of the matrix Δ as follows: setting the i^th row of the matrix Δ to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Δ to the u_imin and the i+1^th row to u_i+1max.
  • 4-4) When the p^th control component and the q^th control component are not a pair of linear constraint components, and each have the paired linear constraint component, determine a key boundary surface of the p-q group according to a following specific method:
  • 4-4-1) The p-q group has three types of rectangular boundary surfaces. Each component of each point on the boundary surfaces of the group has a value that satisfies a following formula. The component paired with the p^th component is denoted as a p′^th component, and the component paired with the q^th component is denoted as a q′^th component.

$\begin{array}{cc}\{\begin{array}{c}{u}_{p\min }\le u_p\le u_{p\max }\\ u_{q\min }\le u_q\le u_{q\max }\\ u_{p^{\prime }}\in \left\{u_{p^{\prime }\min },\frac{u_{p^{\prime }\max }{u}_{p\max }-u_{p^{\prime }\min }{u}_{p\min }-\left(u_{p^{\prime }\max }-u_{p^{\prime }\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}\right\}\\ u_{q^{\prime }}\in \left\{u_{q^{\prime }\min },\frac{u_{q^{\prime }\max }{u}_{q\max }-u_{q^{\prime }\min }{u}_{q\min }-\left(u_{q^{\prime }\max }-u_{q^{\prime }\min }\right)u_q}{\left(u_{q\max }-u_{q\min }\right)}\right\}\\ u_i\in \left\{u_{i\min },u_{i\max }\right\},1\le i\le m,i\ne p,q,p^{\prime },q^{\prime }\\ u_k+u_{k+1}<u_{k\max }+u_{k+1\max },k=1,3,\dots ,2\left[\frac{m}{2}\right]-1,k\ne p,q,p^{\prime },q^{\prime }\end{array}& \left(5\right)\end{array}$

When u_p′=u_p′min and u_q′=u_q′min it is determined that a boundary surface formed by a point satisfying the formula (5) is the type-I rectangular boundary surface.

When u_p′=u_p′min and

$u_{q^{\prime }}=\frac{u_{q^{\prime }\max }{u}_{q\max }-u_{q^{\prime }\min }{u}_{q\min }-\left(u_{q^{\prime }\max }-u_{q^{\prime }\min }\right)u_q}{\left(u_{q\max }-u_{q\min }\right)},$

it is determined that a boundary surface formed by a point satisfying the formula (5) is the type-II rectangular boundary surface.

When

$u_{p^{\prime }}=\frac{u_{p^{\prime }\max }{u}_{p\max }-u_{p^{\prime }\min }{u}_{p\min }-\left(u_{p^{\prime }\max }-u_{p^{\prime }\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}$

and u_q′=u_q′min, is determined that a boundary surface formed by a point satisfying the formula (5) also is the type-II rectangular boundary surface.

When

$u_{p^{\prime }}=\frac{u_{p^{\prime }\max }{u}_{p\max }-u_{p^{\prime }\min }{u}_{p\min }-\left(u_{p^{\prime }\max }-u_{p^{\prime }\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}$ $\mathrm{and}$ $u_{q^{\prime }}=\frac{u_{q^{\prime }\max }{u}_{q\max }-u_{q^{\prime }\min }{u}_{q\min }-\left(u_{q^{\prime }\max }-u_{q^{\prime }\min }\right)u_q}{\left(u_{q\max }-u_{q\min }\right)},$

it is determined that a boundary surface formed by a point satisfying the formula (5) is the type-III rectangular boundary surface.

A vertex of each boundary surface in the group can be represented by an m-row and 4-column matrix, and 4 columns respectively represent vectors corresponding to 4 vertices. A matrix Λ represents a vertex of one type-I rectangular boundary surface, a matrix ¹Π or ²Π represents a vertex of one type-II rectangular boundary surface, and a matrix T represents a vertex of one type-III rectangular boundary surface. Values of the p^th, the p′^th, the q^th, and the q′^th components of four vertices on each boundary surface are shown in p^th, p′^th, q^th, and q′^th rows of the matrix. A value of an i^th component is the u_imax or the u_imin, and an i^th row of the corresponding matrix is the u_imax or the u_imin. However, one pair of linear inequality constraint components cannot simultaneously take a maximum constraint value corresponding to the one pair of linear inequality constraint components, and the maximum constraint value is uniformly represented by an ellipsis in the matrix. For the Λ, the p^th row is (u_pmax, u_pmin, u_pmin, u_pmax), the p′^th row is (u_p′min, u_p′min, u_p′min, u_p′min), the q^th row is (u_qmin, u_qmin, u_qmax, u_qmax), and the q′^th row is (u_q′min, u_q′min, u_q′min, u_q′min).

For the ¹Π, the p^th row is (u_pmax, u_pmin, u_pmin, u_{p max}), the p′ ^th row is (u_p′min, u_p′max, u_p′max, u_p′min), the q^th row is (u_qmin, u_qmin, u_qmax, u_qmax), and the q′^th row is (u_q′min, u_q′min, u_q′min, u_q′min).

For the ²Π, the p^th row is (u_pmax, u_pmin, u_pmin, u_pmax), the p′^th row is (u_p′min, u_p′min, u_p′min, u_p′min), the q^th row is (u_qmin, u_qmin, u_qmax, u_qmax), and the q′^th row is (u_q′max, u_q′max, u_q′min, u_q′min).

For the T, the p^th row is (u_pmax, u_pmin, u_pmin, u_pmax), the p′^th row is (u_p′min, u_p′max, u_p′max, u_p′min), the q^th row is (u_qmin, u_qmin, u_qmax, u_qmax), and the q′^th row is (u_q′max, u_q′max, u_q′min, u_q′min).

$\begin{array}{cc}\Lambda =\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{\prime }\min }& u_{p^{\prime }\min }& u_{p^{\prime }\min }& u_{p^{\prime }\min }\\ \dots & \dots & \dots & \dots \\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{\prime }\min }& u_{q^{\prime }\min }& u_{q^{\prime }\min }& u_{q^{\prime }\min }\\ \dots & \dots & \dots & \dots \end{array}\right],& \left(6\right)\end{array}$ ${ }^1\Pi =\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{\prime }\min }& u_{p^{\prime }\max }& u_{p^{\prime }\max }& u_{p^{\prime }\min }\\ \dots & \dots & \dots & \dots \\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{\prime }\min }& u_{q^{\prime }\min }& u_{q^{\prime }\min }& u_{q^{\prime }\min }\\ \dots & \dots & \dots & \dots \end{array}\right],$ ${ }^2\Pi =\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{\prime }\min }& u_{p^{\prime }\min }& u_{p^{\prime }\min }& u_{p^{\prime }\min }\\ \dots & \dots & \dots & \dots \\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{\prime }\max }& u_{q^{\prime }\max }& u_{q^{\prime }\min }& u_{q^{\prime }\min }\\ \dots & \dots & \dots & \dots \end{array}\right],$ $T=\left[\begin{array}{cccc}\dots & \dots & \dots & \dots \\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{\prime }\min }& u_{p^{\prime }\max }& u_{p^{\prime }\max }& u_{p^{\prime }\min }\\ \dots & \dots & \dots & \dots \\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\\ u_{q^{\prime }\max }& u_{q^{\prime }\max }& u_{q^{\prime }\min }& u_{q^{\prime }\min }\\ \dots & \dots & \dots & \dots \end{array}\right]$

It should be noted that: In the four matrices shown in the above formula, if p is an odd number, the p^th row comes before the p′^th row; or if p is an even number, the p^th row comes after the p′^th row. If q is an odd number, the q^th row comes before the q′^th row; or if q is an even number, the q^th row comes after the q′^th row.

• • 4-4-2) Construct a coordinate rotation transformation G, such that a transformed coordinate axis v₁ is perpendicular to an image of the type-III rectangular boundary surface, and images of all points on the boundary surface have equal coordinate values on the v₁ axis.

$G=\left[\begin{array}{ccc}{g}_{11}& g_{12}& g_{13}\\ g_{21}& g_{22}& g_{23}\\ g_{31}& g_{32}& g_{33}\end{array}\right],B=\left[\begin{array}{ccccccccc}{b}_{11}& \dots & b_{1p}& b_{1p^{\prime }}& \dots & b_{1q}& b_{1q^{\prime }}& \dots & b_{1m}\\ b_{21}& \dots & b_{2p}& b_{2p^{\prime }}& \dots & b_{2q}& b_{2q^{\prime }}& \dots & b_{2m}\\ b_{31}& \dots & b_{3p}& b_{3p^{\prime }}& \dots & b_{3q}& b_{3q^{\prime }}& \dots & b_{3m}\end{array}\right],$ $\mathrm{and}E=\left[\begin{array}{ccccccccc}{e}_{11}& \dots & e_{1p}& e_{1p^{\prime }}& \dots & e_{1q}& e_{1q^{\prime }}& \dots & e_{1m}\\ e_{21}& \dots & e_{2p}& e_{2p^{\prime }}& \dots & e_{2q}& e_{2q^{\prime }}& \dots & e_{2m}\\ e_{31}& \dots & e_{3p}& e_{3p^{\prime }}& \dots & e_{3q}& e_{3q^{\prime }}& \dots & e_{3m}\end{array}\right]$

are set, where g_ij represents an element of an i^th row and a j^th column of the matrix G, and e_ij represents an element of an i^th row and a j^th column of the matrix E.

E=G·B is set, such that

$\begin{array}{cc}{e}_{1p^{\prime }}=\frac{u_{p\max }-u_{p\min }}{u_{p^{\prime }\max }-u_{p^{\prime }\min }}{e}_{1p},& e_{1q^{\prime }}=\frac{u_{q\max }-u_{q\min }}{u_{q^{\prime }\max }-u_{q^{\prime }\min }}{e}_{1q}\end{array}.$

The G and the B are substituted into the E=G·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{g}_{1s}·b_{\mathrm{sp}}=e_{1p}\\ \sum _{s=1}^3{g}_{1s}·b_{{\mathrm{sp}}^{\prime }}=\frac{u_{p\max }-u_{p\min }}{u_{p^{\prime }\max }-u_{p^{\prime }\min }}{e}_{1p}\\ \sum _{s=1}^3{g}_{1s}·b_{\mathrm{sq}}=e_{1q}\\ \sum _{s=1}^3{g}_{1s}·b_{{\mathrm{sq}}^{\prime }}=\frac{u_{q\max }-u_{q\min }}{u_{q^{\prime }\max }-u_{q^{\prime }\min }}{e}_{1q}\end{array}& \left(7\right)\end{array}$

The linear equation set (7) is solved to obtain a first row of the G, and a first row (e₁₁, . . . , e_1m) of the matrix E is calculated according to the E=G·B. If e_1p′>0 and e_1q′22 0, a step 4-4-3) is performed; if e_1p′<0 and e_1q′21 0, a step 4-4-4) is performed; or if e_1p′·e_1q′21 0, a step 4-4-5) is performed.

• • 4-4-3) Process other rows than the rows p, q, p′, q′ of the matrix T according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then perform the step 4-4-5), where i=1, . . . , m, i≠p, p′, q, q′.

The rule is as follows:

• • when e_1i·e_1i+1<0,
  • if e_1i>0 and e_1i+1<0, setting the i^th row of the matrix T to the u_imax and an i+1^th row to u_i+1min; or
  • if e_1i<0 and e_1i+1>0, setting the i^th row of the matrix T to the u_imin and an i+1^th row to u_i+1max;
  • when e_1i<0 and e_1i+1<0,
  • setting the i^th row of the matrix T to the u_imin and an i+1^th row to u_i+1min; or
  • when e_1i<0 and e_1i+1>0,
  • calculating ¹d₀₁=e_1i·u_imax+e_1i+1·u_i+1min and ¹d₀₂=e_1i·u_imin+e_1i+1·u_i+1max and performing determining as follows: if ¹d₀₁>¹d₀₂, setting an i^th row of the matrix T to the u_imax and an i+1^th row to the u_i+1min; if ¹d₀₁<¹z₀₂, setting an i^th row of the matrix T to the u_imin and an i+1^th row to u_i+1max; or if ¹d₀₁=¹d₀₂, setting an i^th row and an i+1^th row of the matrix T as follows: setting the i^th row of the matrix T to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix T to the u_imin and the i+1^th row to u_i+1max.

4-4-4) Process other rows than the rows p, q, p′, q′ of the matrix T according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then perform the step 4-4-5), where i=1, . . . , m, i≠p, p′, q, q′.

The rule is as follows:

• • when e_1i·e_1i+1<0,
  • if e_1i>0 and e_1i+1<0, setting the i^th row of the matrix T to the u_imin and an i+1^th row to u_i+1max; or
  • if e_1i<0 and e_1i+1>0, setting the i^th row of the matrix T to the u_imax and an i+1^th row to u_i+1min;
  • when e_1i<0 and e_1i+1<0,
  • calculating ²d₀₁=²h_1i·u_imax+²h_1i+1·u_i+1min, and ²d₀₂=²h_1i·u_imin+²h_1i+1·u_i+1max; and if ²d₀₁>²d₀₂, setting an i^th row of the matrix T to the u_imin and an i+1^th row to the u_i+1max; if ²d₀₁<²d₀₂, setting an i^th row of the matrix T to the u_imax and an i+1^th row to u_i+1min; or if ²d₀₁=²d₀₂, setting an i^th row and an i+1^th row of the matrix T as follows: setting the i^th row of the matrix T to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix T to the u_imin and the i+1^th row to u_i+1max; or
  • when e_1i>0 and e_1i+1>0;
  • setting the i^th row of the matrix T to the u_imin and an i+1^th row to the u_i+1min.
  • 4-4-5) Construct a coordinate rotation transformation ²T, such that a transformed coordinate axis v₁ is perpendicular to an image of the type-I rectangular boundary surface of the group, where only a first row of the ²T is required because only the v₁ axis is considered.

${ }^{2}T=\left[\begin{array}{ccc}{ }^2{t}_{11}& { }^2{t}_{12}& { }^2{t}_{13}\\ { }^2{t}_{21}& { }^2{t}_{22}& { }^2{t}_{23}\\ { }^2{t}_{31}& { }^2{t}_{32}& { }^2{t}_{33}\end{array}\right],B=\left[\begin{array}{c}{b}_{11}\dots b_{1p}\dots b_{1q}\dots b_{1m}\\ b_{21}\dots b_{2p}\dots b_{2q}\dots b_{2m}\\ b_{31}\dots b_{3p}\dots b_{3q}\dots b_{3m}\end{array}\right],\mathrm{and}$ ${ }^{2}C=\left[\begin{array}{c}{ }^2{c}_{11}\dots { }^2{c}_{1p}\dots { }^2{c}_{1q}\dots { }^2{c}_{1m}\\ { }^2{c}_{21}\dots { }^2{c}_{2p}\dots { }^2{c}_{2q}\dots { }^2{c}_{2m}\\ { }^2{c}_{31}\dots { }^2{c}_{3p}\dots { }^2{c}_{3q}\dots { }^2{c}_{3m}\end{array}\right]$

are set, where ²c_ij represents an element of an i^th row and a j^th column of the matrix ²C, and ²t_ij represents an element of an i^th row and a j^th column of the matrix ²T.

²C=²T·B is set, such that ²c_1p=0, and ²c_1q=0. The ²T and the B are substituted into the ²C=²T·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{ }^2{t}_{1s}·b_{sp}=0\\ \sum _{s=1}^3{ }^2{t}_{1s}·b_{sq}=0\end{array}& \left(8\right)\end{array}$

The linear equation set (8) is solved to obtain the first row (²t₁₁, ²t₁₂,²t₁₃) of the ²T. Then, a first row(²c₁₁, . . . , ²c_1m) of the matrix ²c can be calculated according to the ²C=²T·B.

If ²c_1p′<0 and ²c_1q′<0, a step 4-4-6) is performed; if ²c_1p′>0 and ²c_1q′>0, a step 4-4-7) is performed; or if ²c_1p′·²c_1q′<0, a step 4-4-8) is performed.

4-4-6) Process other rows than the rows p, q, p′, q′ of the matrix Λ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then perform determining as follows:

• • if two key boundary surfaces have been found for the group, perform a step 4-6); otherwise, perform the step 4-4-8).

The rule is as follows:

• • when ²c_1i·²c_1i+1<0,
  • if ²c_1i>0 and ²c_1i+1<0, setting the i^th row of the matrix Λ to the u_imax and an i+1^th row to u_i+1min; or
  • if ²c_1i<0 and ²c_1i+1>0, setting the i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1max;
  • when ²c_1i<0 and ²c_1i+1<0,
  • setting the i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1min; or
  • when ²c_1i>0 and ²c_1i+1>0,
  • calculating ³d₀₁=²c_1i·u_imax+²c_1i+1·u_i+1min, and ³d₀₂=²c_1i·u_imin+²c_1i+1·u_i+1max and if ³d₀₁>³d₀₂, setting an i^th row of the matrix Λ to the u_imax and the i+1^th row to the u_i+1min; if ³d₀₁<³d₀₂, setting an i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1max; or if ³d₀₁=³d₀₂, setting an i^th row and an i+1^th row of the matrix Λ as follows: setting the i^th row of the matrix Λ to the u_imin and the i+1^th row to u_i+1max.
  • 4-4-7) Process other rows than the rows p, q, p′, q′ of the matrix Λ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then perform determining as follows:
  • if two key boundary surfaces have been found for the group, perform a step 4-6); otherwise, perform the step 4-4-8).

The rule is as follows:

• • when ²c_1i·²c_1i+1<0,
  • if ²c_1i>0 and ²c_1i+1<0, setting the i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1max; or
  • if ²c_1i<0 and ²c_1i+1>0, setting the i^th row of the matrix Λ to the u_imax and an i+1^th row to u_i+1min;
  • when ²c_1i<0 and ²c_1i+1<0,
  • calculating ⁴d₀₁=²c_1i·u_imax+²c_1i+1·u_i+1min, and ⁴d₀₂=²c_1i·u_imin+²c_1i+1·u_i+1max and if ⁴d₀₁>⁴d₀₂, setting an i^th row of the matrix Λ to the u_imin and the i+1^th row to the u_i+1max; if ⁴d₀₁<⁴d₀₂, setting an i^th row of the matrix Λ to the u_imax and an i+1^th row to u_i+1min; or if ⁴d₀₁=⁴d₀₂, setting an i^th row and an i+1^th row of the matrix Λ as follows: setting the i^th row of the matrix Λ to the u_imin and the i+1^th row to u_i+1max; or
  • when ²c_1i>0 and ²c_1i+1>0,
  • setting the i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1min.
  • 4-4-8) Construct a coordinate rotation transformation ¹F, such that a transformed coordinate axis is perpendicular v₁ to an image of a boundary surface whose vertex matrix is the ¹Π in the group, and images of all points on the boundary surface have equal coordinate values on the v₁ axis, where only a first row of the ¹F is required because only the axis is considered.

${ }^{1}F=\left[\begin{array}{ccc}{ }^1{f}_{11}& { }^1{f}_{12}& { }^1{f}_{13}\\ { }^1{f}_{21}& { }^1{f}_{22}& { }^1{f}_{23}\\ { }^1{f}_{31}& { }^1{f}_{32}& { }^1{f}_{33}\end{array}\right],B=\left[\begin{array}{cccccccc}{b}_{11}& \dots & b_{1p}& b_{1p^{\prime }}& \dots & b_{1q}& \dots & b_{1m}\\ b_{21}& \dots & b_{2p}& b_{2p^{\prime }}& \dots & b_{2q}& \dots & b_{2m}\\ b_{31}& \dots & b_{3p}& b_{3p^{\prime }}& \dots & b_{3q}& \dots & b_{3m}\end{array}\right],$ $\mathrm{and}{ }^{1}H=\left[\begin{array}{cccccccc}{ }^1{h}_{11}& \dots & { }^1{h}_{1p}& { }^1{h}_{1p^{\prime }}& \dots & { }^1{h}_{1q}& \dots & { }^1{h}_{1m}\\ { }^1{h}_{21}& \dots & { }^1{h}_{2p}& { }^1{h}_{2p^{\prime }}& \dots & { }^1{h}_{2q}& \dots & { }^1{h}_{2m}\\ { }^1{h}_{31}& \dots & { }^1{h}_{3p}& { }^1{h}_{3p^{\prime }}& \dots & { }^1{h}_{3q}& \dots & { }^1{h}_{3m}\end{array}\right]$

are set, where ¹f_ij represents an element of an i^th row and a j^th column of the matrix ¹F, and ¹h_ij represents an element of an i^th row and a j^th column of the matrix ¹H.

¹H=¹F·B is set, such that

${ }^1{h}_{1p^{\prime }}=\frac{u_{p\max }-u_{p\min }}{u_{p^{\prime }\max }-u_{p^{\prime }\min }}{ }^1{h}_{1p},$

and ¹h_1q=0. The ¹F and the B are substituted into the ¹H=¹F·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{ }^1{f}_{1s}·b_{sp}={ }^1{h}_{1p}\\ \sum _{s=1}^3{ }^1{f}_{1s}·b_{{\mathrm{sp}}^{\prime }}=\frac{u_{p\max }-u_{p\min }}{u_{p^{\prime }\max }-u_{p^{\prime }\min }}{ }^1{h}_{1p}\\ \sum _{s=1}^3{ }^1{f}_{1s}·b_{sq}=0\end{array}& \left(9\right)\end{array}$

The linear equation set (9) is solved to obtain the first row of the ¹F, and a first row (¹h₁₁, . . . , ¹h_1m) of the matrix ¹H according to the ¹H=¹F·B.

If ¹h_1p′>0, and ¹h_1p′<0, a step 4-4-9) is performed; if ¹h_1p′<0 and ¹h_1p′<0, a step 4-4-10) is performed; or if ¹h_1p′·¹h_1p′>0, a step 4-4-11) is performed.

4-4-9) Process other rows than the rows p, q, p′, q′ of the matrix ¹Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then perform determining as follows:

• • if two key boundary surfaces have been found for the group, perform a step 4-6); otherwise, perform the step 4-4-11).

The rule is as follows:

• • when ¹h_1i·¹h_1i+1<0,
  • if ¹h_1i>0 and ¹h_1i+1<0, setting the i^th row of the matrix ¹Π to the u_imax and an i+1^th row to u_i+1min; or
  • if ¹h_1i<0 and ¹h_1i+1>0, setting the i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1max;
  • when ¹h_1i<0 and ¹h_1i+1<0,
  • setting the i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1min, or
  • when ¹h_1i>0 and ¹h_1i+1>0,
  • calculating ⁵d₀₁=¹h_1i·u_imax+¹h_1i+1·u_i+1min, and ⁵d₀₂=¹h_1i·u_imin+¹h_1i+1·u_i+1max; and if ⁵d₀₁>⁵d₀₂, setting an i^th row of the matrix ¹Π to the u_imax and the i+1^th row to the u_i+1min; if ⁵d₀₁<⁵d₀₂, setting an i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1max; or if ⁵d₀₁=⁵d₀₂, setting an i^th row and an i+1^th row of the matrix ¹Π as follows: setting the i^th row of the matrix ¹Π to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix ¹Π to the u_imin and i+1^th row to u_i+1max.

4-4-10) Process other rows than the rows p, q, p′, q′ of the matrix ¹Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then perform determining as follows:

if two key boundary surfaces have been found for the group, perform a step 4-6); otherwise, perform the step 4-4-11).

The rule is as follows:

• • when ¹h_1i·¹h_1i+1<0,
  • if ¹h_1i>0 and ¹h_1i+1<0, setting the i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1max; or
  • if ¹h_1i<0 and ¹h_1i+1>0, setting the i^th row of the matrix ¹Π to the u_imax and an i+1^th row to u_i+1min;
  • when ¹h_1i>0 and ¹h_1i+1>0,
  • setting the i^th row of the matrix ¹Π to the u_imin and an i+1^th row to u_i+1min, or
  • when ¹h_1i<0 and ¹h_1i+1<0,
  • calculating ⁶d₀₁=¹h_1i·u_imax+¹h_1i+1·u_i+1min, and ⁶d₀₂=¹h_1i·u_imin+¹h_1i+1·u_i+1max; and if ⁶d₀₁>⁶d₀₂, setting an i^th row of the matrix ¹Π to the u_imin and the i+1^th row to the u_i+1max; if ⁶d₀₁<⁶d₀₂, setting an i^th row of the matrix ¹Π to the u_imax and an i+1^th row to u_i+1min; or if ⁶d₀₁=⁶d₀₂, setting an i^th row and an i+1^th row of the matrix ¹Π as follows: setting the i^th row of the matrix ¹Π to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix ¹Π to the u_imin and i+1^th row to u_i+1max.

4-4-11) Construct a coordinate rotation transformation ²F, such that a transformed coordinate axis v₁ is perpendicular to an image of a boundary surface whose vertex matrix is the ²Λ in the group, and images of all points on the boundary surface have equal coordinate values on the v₁ axis, where only a first row of the ²F is required because only the v₁ axis is considered.

${ }^{2}F=\left[\begin{array}{ccc}{ }^2{f}_{11}& { }^2{f}_{12}& { }^2{f}_{13}\\ { }^2{f}_{21}& { }^2{f}_{22}& { }^2{f}_{23}\\ { }^2{f}_{31}& { }^2{f}_{32}& { }^2{f}_{33}\end{array}\right],B=\left[\begin{array}{cccccccc}{b}_{11}& \dots & b_{1p}& b_{1p^{\prime }}& \dots & b_{1q}& \dots & b_{1m}\\ b_{21}& \dots & b_{2p}& b_{2p^{\prime }}& \dots & b_{2q}& \dots & b_{2m}\\ b_{31}& \dots & b_{3p}& b_{3p^{\prime }}& \dots & b_{3q}& \dots & b_{3m}\end{array}\right],$ $\mathrm{and}{ }^{2}H=\left[\begin{array}{cccccccc}{ }^2{h}_{11}& \dots & { }^2{h}_{1p}& { }^2{h}_{1p^{\prime }}& \dots & { }^2{h}_{1q}& \dots & { }^2{h}_{1m}\\ { }^2{h}_{21}& \dots & { }^2{h}_{2p}& { }^2{h}_{2p^{\prime }}& \dots & { }^2{h}_{2q}& \dots & { }^2{h}_{2m}\\ { }^2{h}_{31}& \dots & { }^2{h}_{3p}& { }^2{h}_{3p^{\prime }}& \dots & { }^2{h}_{3q}& \dots & { }^2{h}_{3m}\end{array}\right]$

are set, where ²f_ij represents an element of an i^th row and a j^th column of the matrix ²F, and ²h_ij represents an element of an i^th row and a j^th column of the matrix ²H.

²H=²F·B is set, such that

${ }^2{h}_{1q^{\prime }}=\frac{u_{q\max }-u_{q\min }}{u_{q\max }-u_{q\min }}{ }^2{h}_{1q},$

and ²h_1p=0. The ²F and the B are substituted into the ²H=²F·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{cc}\sum _{s=1}^3{ }^2{f}_{1s}·b_{\mathrm{sq}}& =0\\ \sum _{s=1}^3{ }^2{f}_{1s}·b_{\mathrm{sq}}& ={ }^2{h}_{1q}\\ \sum _{s=1}^3{ }^2{f}_{1s}·b_{{\mathrm{sq}}^{\prime }}& =\frac{u_{q\max }-u_{q\min }}{u_{q^{\prime }\max }-u_{q^{\prime }\min }}{ }^2{h}_{1q}\end{array}& \left(10\right)\end{array}$

The linear equation set (10) is solved to obtain the first row of the ²F, and a first row (²h₁₁, . . . , ²h_1m) of the matrix ²H according to the ²H=²F·B. If and ²h_1p′<0 and ²h_1q′>0, a step 4-4-12) is performed; if ²h_1p′>0 and ²h_1q′<0, a step 4-4-13) is performed; or if ²h_1p′·²h_1q′>0, a step 4-6) is performed.

4-4-12) Process other rows than the rows p, q, p′, q′ of the matrix ¹Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then perform the step 4-6).

The rule is as follows:

• • when ²h_1i·²h_1i+1<0,
  • if ²h_1i>0 and ²h_1i+1<0, setting the i^th row of the matrix ²Π to the u_imax and an i+1^th row to u_i+1min; or
  • if ²h_1i<0 and ²h_1i+1>0, setting the i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1max;
  • when ²h_1i<0 and ²h_1i+1<0,
  • setting the i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1min, or
  • when ²h_1i<0 and ²h_1i+1<0,
  • calculating ⁷d₀₁=²h_1i·u_imax+²h_1i+1·u_i+1min, and ⁷d₀₂=²h_1i·u_imin+²h_1i+1·u_i+1max; and if ⁷d₀₁>⁷d₀₂, setting an i^th row of the matrix ²Π to the u_imax and the i+1^th row to the u_i+1min; if ⁷d₀₁<⁷d₀₂, setting an i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1max; or if ⁷d₀₁=⁷d₀₂, setting an i^th row and an i+1^th row of the matrix ²Π as follows: setting the i^th row of the matrix ²Π to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix ²Π to the u_imin and i+1^th row to u_i+1max.

4-4-13) Process other rows than the rows p, q, p′, q′ of the matrix ²Π according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω, and i=1, . . . , m, i≠p, p′, q, q′; and then perform the step 4-6).

The rule is as follows:

• • when ²h_1i·²h_1i+1<0,
  • if ²h_1i>0 and ²h_1i+1<0, setting the i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1max; or
  • if ²h_1i<0 and ²h_1i+1>0, setting the i^th row of the matrix ¹Π to the u_imax and an i+1^th row to u_i+1min;
  • when ²h_1i>0 and ²h_1i+1>0,
  • calculating ⁸d₀₁=²h_1i·u_imax+²h_1i+1·u_i+1min, and ⁸d₀₂=²h_1i·u_imin+²h_1i+1·u_i+1max; and if ⁸d₀₁>⁸d₀₂, setting an i^th row of the matrix ²Π to the u_imin and the i+1^th row to the u_i+1max; if ⁸d₀₁<⁸d₀₂, setting an i^th row of the matrix ²Π to the u_imax and an i+1^th row to u_i+1min; or if ⁸d₀₁=⁸d₀₂, setting an i^th row and an i+1^th row of the matrix ²Π as follows: setting the i^th row of the matrix ²Π to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix ²Π to the u_imin and i+1^th row to u_i+1max; or
  • when ²h_1i>0 and ²h_1i+1>0,
  • setting the i^th row of the matrix ²Π to the u_imin and an i+1^th row to u_i+1min.

4-5) When the p^th control component and the q^th control component are not a pair of linear constraint components, and the q^th control component has no paired linear constraint component, determine a key boundary surface of the p-q group according to a following specific method:

4-5-1) This situation occurs only when m is an odd number and q=m. The p-q group contains both the type-I and the type-II rectangular boundary surfaces. Each component of each point on the boundary surfaces of the group has a value that satisfies a following formula (11). A component paired with the p^th component is denoted as a p′^th component.

$\begin{array}{cc}\{\begin{array}{c}{u}_{\mathrm{pmin}}\le u_p\le u_{p\max }\\ u_{q\min }\le u_q\le u_{\mathrm{qmax}}\\ u_{p^{\prime }}\in \left\{u_{p^{\prime }\min },\frac{u_{p^{\prime }\max }{u}_{p\max }-u_{p^{\prime }\min }{u}_{p\min }-\left(u_{p^{\prime }\max }-u_{p^{\prime }\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)}\right\}\\ u_i\in \left\{u_{i\min },u_{i\max }\right\},1\le i<m,i\ne p,q,p^{\prime }\\ u_k+u_{k+1}<u_{k\max }+u_{k+1\max },k=1,3,\dots ,2\left[\frac{m}{2}\right]-1,k\ne p,q,p^{\prime }\end{array}& \left(11\right)\end{array}$

When u_p′=u_p′min, it is determined that a boundary surface formed by a point satisfying the formula (11) is the type-I rectangular boundary surface; or when

$u_{p^{\prime }}=\frac{u_{p^{\prime }\max }{u}_{p\max }-u_{p^{\prime }\min }{u}_{p\min }-\left(u_{p^{\prime }\max }-u_{p^{\prime }\min }\right)u_p}{\left(u_{p\max }-u_{p\min }\right)},$

it is determined that a boundary surface formed by a point satisfying the formula (11) is the type-II rectangular boundary surface.

A vertex of each boundary surface in the group can be represented by an m row and 4-column matrix, and 4 columns respectively represent vectors corresponding to 4 vertices. A matrix Λ′ is used to represent a vertex of one type-I rectangular boundary surface, and a matrix Π′ is used to represent a vertex of one type-I rectangular boundary surface. Values of the p^th p′^th, and q^th components of four vertices on each boundary surface are shown in p^th, p′^th and q^th rows of the matrix. A value of an i(i=1, . . . , m, i≠p, p′, q)^th component is the u_imax or the u_min, and an i(i=1, . . . , m, i≠p, p′, q)^th row of the corresponding matrix is the u_imax or the u_imin. However, one pair of linear constraint components cannot simultaneously take a maximum constraint value corresponding to the one pair of linear constraint components, and the maximum constraint value is uniformly represented by an ellipsis in the matrix. For the Λ′, the p^th row is (u_pmax, u_pmin, u_pmin, u_pmax), the p′^th row is (u_p′min, u_p′min, u_p′min, u_p′min), the q^th row is) (u_qmin, u_qmin, u_qmax, u_qmax), and the q^th row is (u_q′min, u_q′min, u_q′min, u_q′min).

${\Lambda }^{\prime }=\left[\begin{array}{cccc}\cdots & \cdots & \cdots & \cdots \\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{\prime }\min }& u_{p^{\prime }\min }& u_{p^{\prime }\min }& u_{p^{\prime }\min }\\ \cdots & \cdots & \cdots & \cdots \\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\end{array}\right],{\prod }^{\prime }=\left[\begin{array}{cccc}\cdots & \cdots & \cdots & \cdots \\ u_{p\max }& u_{p\min }& u_{p\min }& u_{p\max }\\ u_{p^{\prime }\min }& u_{p^{\prime }\max }& u_{p^{\prime }\max }& u_{p^{\prime }\min }\\ \cdots & \cdots & \cdots & \cdots \\ u_{q\min }& u_{q\min }& u_{q\max }& u_{q\max }\end{array}\right]$

In the matrices shown in the above formula, if P is an odd number, the p^th row comes before the p′^th row; or if P is an even number, the p^th row comes after the p′^th row.

4-5-2) Construct a coordinate rotation transformation ³T, such that a transformed coordinate axis v₁ is perpendicular to an image of the type-I rectangular boundary surface of the group, where only a first row of the ³T is required because only the v₁ axis is considered.

${ }^{3}T=\left[\begin{array}{ccc}{ }^3{t}_{11}& { }^3{t}_{12}& { }^3{t}_{13}\\ { }^3{t}_{21}& { }^3{t}_{22}& { }^3{t}_{23}\\ { }^3{t}_{31}& { }^3{t}_{32}& { }^3{t}_{33}\end{array}\right],B=\left[\begin{array}{ccccc}{b}_{11}& \dots & b_{1p}& \dots & b_{1q}\\ b_{21}& \dots & b_{2p}& \dots & b_{2q}\\ b_{31}& \dots & b_{3p}& \dots & b_{3q}\end{array}\right],\mathrm{and}{ }^{3}C=\left[\begin{array}{ccccc}{ }^3{c}_{11}& \dots & { }^3{c}_{1p}& \dots & { }^3{c}_{1q}\\ { }^3{c}_{21}& \dots & { }^3{c}_{2p}& \dots & { }^3{c}_{2q}\\ { }^3{c}_{31}& \dots & { }^3{c}_{3p}& \dots & { }^3{c}_{3q}\end{array}\right]$

are set, where ³t_ij represents an element of an i^th row and a j^th column of the matrix ³T, and ³c_ij represents an element of an i^th row and a j^th column of the matrix ³C.

³C=³T·B is set, such that ³c_1p=0, and ³c_1q=0. The ³T and the B are substituted into the ³C=³T·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{cc}\sum _{s=1}^3{ }^3{t}_{1s}·b_{\mathrm{sp}}& =0\\ \sum _{s=1}^3{ }^3{t}_{1s}·b_{\mathrm{sq}}& =0\end{array}& \left(12\right)\end{array}$

The linear equation set (12) is solved to obtain the first row (³t₁₁, ³t₁₂, ³t₁₃) of the ³T. Then, a first row (³c₁₁, . . . , ³c_1m) of the matrix ³C can be calculated according to the ³C=³T·B. If ³c_1p′<0, a step 4-5-3) is performed; or if ³c_1p′>0, a step 4-5-4) is performed.

4-5-3) Process other rows than the rows p, q, p′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then perform a step 4-5-5), where i=1, . . . , m, i≠p, p′, q.

The rule is as follows:

• • when ³c_1i·³c_1i+1<0,
  • if ³c_1i>0 and ³c_1i+1<0, setting the i^th row of the matrix Λ′ to the u_imax and an i+1^th row to u_i+1min; or
  • if ³c_1i<0 and ³c_1i+1>0, setting the i^th row of the matrix Λ′ to the u_imin and an i+1^th row to u_i+1max;
  • when ³c_1i<0 and ³c_1i+1<0,
  • setting the i^th row of the matrix Λ′ to the u_imin and an i+1^th row to u_i+1min; or
  • when ³c_1i>0 and ³c_1i+1>0,
  • calculating ⁹d₀₁=³c_1i·u_imax+³c_1i+1·u_i+1min, and ⁹d₀₂=³c_1i·u_imin+³c_1i+1·u_i+1max; and
  • if ⁹d₀₁>⁹d₀₂, setting an i^th row of the matrix Λ′ to the u_imax and the i+1^th row to u_i+1min; if ⁹d₀₁<⁹d₀₂, setting an i^th row of the matrix Λ to the u_imin and an i+1^th row to u_i+1max; or if ⁹d₀₁=⁹d₀₂, setting an i^th row and an i+1^th row of the matrix Λ′ as follows: setting the i^th row of the matrix Λ′ to the u_imin and the i+1^th row to u_i+1max.

4-5-4) Process other rows than the rows p, q, p′ of the matrix Λ′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then perform a step 4-5-5), where i=1, . . . , m, i≠p, p′, q.

The rule is as follows:

• • when ³c_1i·³c_1i+1<0,
  • if ³c_1i>0 and ³c_1i+1<0, setting the i^th row of the matrix Λ′ to the u_imin and an i+1^th row to u_i+1max; or
  • if ³c_1i<0 and ³c_1i+1>0, setting the i^th row of the matrix Λ′ to the u_imax and an i+1^th row to u_i+1min;
  • when ³c_1i>0 and ³c_1i+1>0,
  • setting the i^th row of the matrix Λ′ to the u_imin and an i+1^th row to u_i+1min; or
  • when ³c_1i<0 and ³c_1i+1<0,
  • calculating ¹⁰d₀₁=³c_1i·u_imax+³c_1i+1·u_i+1min, and ¹⁰d₀₂=³c_1i·u_imin+³c_1i+1·u_i+1max; and if ¹⁰d₀₁>¹⁰d₀₂, setting an i^th row of the matrix Λ′ to the u_imin and the i+1^th row to u_i+1max; if ¹⁰d₀₁<¹⁰d₀₂, setting an i^th row of the matrix Λ to the u_imax and an i+1^th row to u_i+1min; or if ¹⁰d₀₁=¹⁰d₀₂, setting an i^th row and an i+1^th row of the matrix Λ′ as follows: setting the i^th row of the matrix Λ′ to the u_imin and the i+1^th row to u_i+1max.

4-5-5) Construct a coordinate rotation transformation ³F, such that a transformed coordinate axis v₁ is perpendicular to an image of a boundary surface whose vertex matrix is the Π′ in the group, and images of all points on the boundary surface have equal coordinate values on the v₁ axis, where only a first row of the ³F is required because only the v₁ axis is considered.

${ }^{3}F=\left[\begin{array}{ccc}{ }^3{f}_{11}& { }^3{f}_{12}& { }^3{f}_{13}\\ { }^3{f}_{21}& { }^3{f}_{22}& { }^3{f}_{23}\\ { }^3{f}_{31}& { }^3{f}_{32}& { }^3{f}_{33}\end{array}\right],B=\left[\begin{array}{cccccc}{b}_{11}& \dots & b_{1p}& b_{1p^{\prime }}& \dots & b_{1q}\\ b_{21}& \dots & b_{2p}& b_{2p^{\prime }}& \dots & b_{2q}\\ b_{31}& \dots & b_{3p}& b_{3p^{\prime }}& \dots & b_{3q}\end{array}\right],\mathrm{and}{ }^{3}H=\left[\begin{array}{cccccc}{ }^3{h}_{11}& \dots & { }^3{h}_{1p}& { }^3{h}_{1p^{\prime }}& \dots & { }^3{h}_{1q}\\ { }^3{h}_{21}& \dots & { }^3{h}_{2p}& { }^3{h}_{2p^{\prime }}& \dots & { }^3{h}_{2q}\\ { }^3{h}_{31}& \dots & { }^3{h}_{3p}& { }^3{h}_{3p^{\prime }}& \dots & { }^3{h}_{3q}\end{array}\right]$

are set, where ³h_ij represents an element of an i^th row and a j^th column of the matrix ³H, and ³f_ij represents an element of an i^th row and a j^th column of the matrix ³F.

³H=³F·B is set, such that

${ }^3{h}_{1p^{\prime }}=\frac{u_{p\max }-u_{p\min }}{u_{p^{\prime }\max }-u_{p^{\prime }\min }}{ }^3{h}_{1p},$

and ³h_1q=0. The ³F and the B are substituted into the ³H=³F·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{cc}\sum _{s=1}^3{ }^3{f}_{1s}·b_{\mathrm{sp}}& {=}^3{h}_{1p}\\ \sum _{s=1}^3{ }^3{f}_{1s}·b_{sp}& =\frac{u_{\mathrm{pmax}}-u_{\mathrm{pmin}}}{u_{\mathrm{pmax}}-u_{\mathrm{pmin}}}{ }^3{h}_{1p}\\ \sum _{s=1}^3{ }^3{f}_{1s}·b_{\mathrm{sq}}& =0\end{array}& \left(13\right)\end{array}$

The linear equation set (13) is solved to obtain the first row of the ³F, and a first row (³h₁₁, . . . , ³h_1m) of the matrix ³H according to the ³H=³F·B. If ³h_1p′>0, a step 4-5-6) is performed; or if ³h_1p′<0, a step 4-5-7) is performed.

4-5-6) Process other rows than the rows p, q, p′ of the matrix Π′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then perform a step 4-6), where i=1, . . . , m, i≠p, p′, q.

The rule is as follows:

• • when ³h_1i·³h_1i+1<0,
  • if ³h_1i>0 and ³h_1i+1<0, setting the i^th row of the matrix Π to the u_imax and an i+1^th row to u_i+1min; or
  • if ³h_1i<0 and ³h_1i+1>0, setting the i^th row of the matrix Π to the u_imin and an i+1^th row to u_i+1max;
  • when ³h_1i<0 and ³h_1i+1<0,
  • setting the i^th row of the matrix Π to the u_imin and an i+1^th row to u_i+1min, or
  • when ³h_1i>0 and ³h_1i+1>0,
  • calculating ¹¹d₀₁=³h_1i·u_imax+³h_1i+1·u_i+1min, and ¹¹d₀₂=³h_1i·u_imin+³h_1i+1·u_i+1max; and if ¹¹d₀₁>¹¹d₀₂, setting an i^th row of the matrix Π to the u_imax and the i+1^th row to the u_i+1min; if ¹¹d₀₁<¹¹d₀₂, setting an i^th row of the matrix Π to the u_imin and an i+1^th row to u_i+1max; or if ¹¹d₀₁=¹¹d₀₂, setting an i^th row and an i+1^th row of the matrix Π as follows: setting the i^th row of the matrix Π to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Π to the u_imin and i+1^th row to u_i+1max.

4-5-7) Process other rows than the rows p, q, p′ of the matrix Π′ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then perform a step 4-6), where i=1, . . . , m, i≠p, p′, q.

The rule is as follows:

• • when ³h_1i·³h_1i+1<0,
  • if ³h_1i>0 and ³h_1i+1<0, setting the i^th row of the matrix Π to the u_imin and an i+1^th row to u_i+1max; or
  • if ³h_1i<0 and ³h_1i+1>0, setting the i^th row of the matrix Π to the u_imax and an i+1^th row to u_i+1min;
  • when ³h_1i<0 and ³h_1i+1<0,
  • calculating ¹²d₀₁=³h_1i·u_imax+³h_1i+1·u_i+1min, and ¹²d₀₂=³h_1i·u_imin+³h_1i+1·u_i+1max; and if ¹²d₀₁>¹²d₀₂, setting an i^th row of the matrix Π to the u_imin and the i+1^th row to the u_i+1max; if ¹²d₀₁<¹²d₀₂, setting an i^th row of the matrix Π to the u_imax and an i+1^th row to u_i+1min; or if ¹²d₀₁=¹²d₀₂, setting an i^th row and an i+1^th row of the matrix Π as follows: setting the i^th row of the matrix Π to the u_imax and the i+1^th row to u_i+1min; or setting the i^th row of the matrix Π′ to the u_imin and i+1^th row to u_i+1max; or
  • when ³h_1i>0 and ³h_1i+1>0,
  • setting the i^th row of the matrix Π′ to the u_imin and an i+1^th row to u_i+1min.

4-6) Return to the step 4-1), select a next group for which no key boundary surface has been determined, until the key boundary surface has been determined for all groups, and then perform a step 5).

5) Check all key boundary surfaces in the Ω, and perform determining as follows: if there are identical key boundary surfaces, retain only one of the identical key boundary surfaces, and remove other key boundary surfaces that are the same as the retained one key boundary surface; after processing all the key boundary surfaces, form a new set of key boundary surfaces, which is denoted as Ω′; and then perform a step 6).

6) All key boundary surfaces in the Ω′ are all key boundary surfaces in the control set. All vertices of each key boundary surface in the Ω′ are mapped according to v=B·u_ELR to obtain all vertices of a boundary surface corresponding to the control attainable set, so as to determine the boundary surface of the control attainable set, where the boundary surface of the control attainable set is a quadrilateral or triangle. Boundary surfaces that are of the control attainable set and determined based on all the key boundary surfaces in the Ω′ constitute the boundary ∂(Φ_ELR) of the control attainable set.

The method in the present disclosure is further described in detail below with reference to a specific embodiment.

EMBODIMENT

In this embodiment, a control attainable set of a four-wheel independently driven and independently steered vehicle is calculated.

Given that four longitudinal forces of four wheels are F_L1, F_L2, F_L3, and F_L4, respectively, and four lateral forces are F_T1, F_T2, F_T3, and F_T4 respectively. It is set that u_v=(F_L1, F_T1, F_L2, F_T2, F_L3,F_T3, F_L4, F_T4)^T, F_Limin≤F_Li≤F_Limax, F_Timin≤F_Ti≤F_Timax, i=1, . . . , 4, F_Limin, F_Timin respectively represent minimum values of the longitudinal and lateral forces of each wheel, F_Limax, F_Timax respectively represent maximum values of the longitudinal and lateral forces of each wheel, [F_L1min, . . . , F_L4min]=[−16, −16, −19.3, −19.3], [F_T1min, . . . , F_T4min]=[−5, −5, −9.21, −9.21], [F_L1max, . . . , F_L4max]=[16, 16, 9.3, 9.3], and [F_T1max, . . . , F_T4max]=[5,5,9.21,9.21]. There is a following linear relationship between an i^th longitudinal force and an i^th lateral force: (F_Limax-F_Limin)F_Ti+(F_Timax-F_Timax)F_Li≤F_TimaxF_Limax−F_TiminF_Limin. All values of the u_v constitute a control set Ω_v={u_v} of the four-wheel independently driven and independently steered vehicle. In addition, a wheel force control efficiency matrix B_v is known:

$B_v=\left(\begin{array}{cccccccc}0.9397& -0.342& 0.9563& -0.2924& 0.9659& 0.2588& 0.9744& 0.2249\\ -0.342& 0.9397& -0.2924& 0.9563& 0.2588& 0.9659& 0.2249& 0.9744\\ 1.0762& 0.1790& 1.5435& -1.0787& -1.3403& -0.9867& -1.0118& 0.4121\end{array}\right),$

The control attainable set Φ_v of the four-wheel independently driven and independently steered vehicle is calculated according to a following formula: Φ_v={v_VS|v_VS=B_v·u_VS=(F_LS, F_TS, M_S)^T, u_vs∈Ω_v}, where F_LS represents a specific overall longitudinal force of the vehicle, F_TS represents a specific overall lateral force of the vehicle, and M_s represents a specific overall yawing moment of the vehicle.

This embodiment proposes a method for calculating a control attainable set of a redundant drive system under a linear constraint, including following steps:

1) Construct a control attainable set of a redundant drive system under each pair of linear constraint control components.

In this embodiment, a correspondence between each physical quantity and the terms used in the foregoing description is as follows: Ω_v corresponds to Ω_ELR, B_v corresponds to B, Φ_v corresponds to Φ_ELR, and u_v=(F_L1, F_T1, F_L2, F_T2, F_L3, F_T3, F_L4, F_T4)^T corresponds to u_ELR=(u₁, . . . , u₈)^T The physical problem is to:

• • determine a control containable set of an 8-dimensional control vector u_ELR=(u₁, . . . , u₈)^T, where u₁ and u₂ are a pair of control components, u₃ and u₄ are a pair of control components, u₅ and u₆ are a pair of control components, and u₇ and u₈ are a pair of control components; and each control component u_i has a value range, namely u_imin≤u_i≤u_imax, where i=1, . . . , 8,

$\left[u_{1\min },\dots ,u_{8\min }\right]=\left[-16,-5,-16,-5,-193,-921,-193,-921\right],\mathrm{and}$ $\left[u_{1\max },\dots ,u_{8\max }\right]=\left[16,5,16,5,19.3,9.21,19.3,9.21\right].$

There is a following linear relationship between each pair of control components:

$\left(u_{k\max }-u_{k\min }\right)u_{k+1}+\left(u_{k+1\max }-u_{k+1\min }\right)u_k\le u_{k+1\max }{u}_{k\max }-u_{k+1\min }{u}_{k\min },k=1,3,5,7.$

Ω_ELR={u_ELR} is set, where

$u_{\mathrm{ELR}}={\left(u_1,\dots ,u_8\right)}^T$ $s.t.$ $u_{\mathrm{imin}}\le u_i\le u_{\mathrm{imax}},i=1,L,8;$ $\left(u_{k\max }-u_{k\min }\right)u_{k+1}+\left(u_{k+1\max }-u_{k+1\min }\right)u_k\le u_{k+1\max }{u}_{k\max }-u_{k+1\min }{u}_{k\min },k=1,3,5,7;$

Each control vector u_ELR of the Ω_ELR generates a control attainable vector v through following mapping: v=B·u_ELR, where B:R⁸→R³,

$B=\left(\begin{array}{cccccccc}0.9397& -0.342& 0.9563& -0.2924& 0.9659& 0.2588& 0.9744& 0.2249\\ -0.342& 0.9397& -0.2924& 0.9563& 0.2588& 0.9659& 0.2249& 0.9744\\ 1.0762& 0.179& 1.5435& -1.0787& -1.3403& -0.9867& -1.0118& 0.4121\end{array}\right).$

Requirement: Determine ∂(Φ_ELR) of the control attainable set.

2) Classify all boundary surfaces of the control set Ω_ELR into four types.

A point obtained by taking a corresponding maximum or minimum constraint value of each component u_i(i=1, . . . , 8) of the control vector u_ELR=(u₁, . . . , u₈)^T is a vertex of the control set Ω_ELR. The 8-dimensional control set Ω_ELR in this embodiment has 3⁴ vertices. Boundary surfaces constituting ∂(Ω_ELR) are a type-rectangular boundary surface, a type-rectangular boundary surface, a type−=2\*ROMAN I rectangular boundary surface, and a triangular boundary surface.

3) Divide all the boundary surfaces of the control set into C₈² groups.

Values of a p^th control component u_p, u_q and a q^th control component (1≤p≤8, 1≤q≤8, p<q) are between the corresponding minimum and maximum values, and values of other 6 components u_i(1≤i≤8, i≠p, i≠q) are the corresponding minimum or maximum value, namely, u_i=u_imax or u_imin Through such combination, a total of 2⁶ boundary surfaces are obtained and added to a group referred to as a p-q group. All the boundary surfaces of the control set are divided into a total of 28 groups, namely, a 1-2 group, a 1-3 group, . . . , and a 7-8 group.

4) Determine a key boundary surface for each of the C₈² groups in the step 3).

There are a total of 28 groups. Due to limited space, only the 1-2 group and the 3-5 group are used as examples to illustrate a process of determining the key boundary surface. A set of key boundary surfaces is denoted as Ω, and the Ω is initialized as an empty set.

4-1) Denote the 1-2 group as a current p-q group.

4-2) Perform a step 4-3) because a first control component and a second control component are a pair of linear constraint components.

4-3)

4-3-1) Boundary surfaces of the group are all triangular boundary surfaces.

4-3-2) Construct a coordinate rotation transformation ¹T, such that a transformed coordinate axis v₁ is perpendicular to an image of the triangular boundary surface of the group, where only a first row of the ¹T is required because only the v₁ axis is considered.

Given that

$B=\left(\begin{array}{cccccccc}0.9397& -0.342& 0.9563& -0.2924& 0.9659& 0.2588& 0.9744& 0.2249\\ -0.342& 0.9397& -0.2924& 0.9563& 0.2588& 0.9659& 0.2249& 0.9744\\ 1.0762& 0.179& 1.5435& -1.0787& -1.3403& -0.9867& -1.0118& 0.4121\end{array}\right),1^{}T=\left[\begin{array}{ccc}{1}^{}{t}_{11}& 1^{}{t}_{12}& 1^{}{t}_{13}\\ 1^{}{t}_{21}& 1^{}{t}_{22}& 1^{}{t}_{23}\\ 1^{}{t}_{31}& 1^{}{t}_{32}& 1^{}{t}_{33}\end{array}\right],{\mathrm{and}}^{1}C=\left[\begin{array}{cccc}{1}^{}{c}_{11}& 1^{}{c}_{12}& \dots & 1^{}{c}_{18}\\ 1^{}{c}_{21}& 1^{}{c}_{22}& \dots & 1^{}{c}_{28}\\ 1^{}{c}_{31}& 1^{}{c}_{32}& \dots & 1^{}{c}_{38}\end{array}\right]$

are set. ¹C=¹T·B is set, such that ¹C₁₁=0, and ¹C₁₂=0.

The ¹T and the B are substituted into the ¹C=¹T·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{1}^{}{t}_{1s}·b_{s1}=0\\ \sum _{s=1}^3{1}^{}{t}_{1s}·b_{s2}=0\end{array}& \left(14\right)\end{array}$

The linear equation set (14) is solved. Set ¹t₁₃=1, such that ¹t₁₁=−1.4, ¹t₁₂=−0.7 can be calculated.

Then, the first row of the matrix ¹C can be calculated according to the ¹C=¹T·B, which is (¹c₁₁, . . . , ¹C₁₈)=(0, 0, 0.4093, −1.3388, −2.8738, −2.0252, −2.5334, −0.5849) Then, a step 4-3-3) is performed.

4-3-3) Due to ¹c₁₃·¹c₁₄<0, and ¹c₁₃>0, and ¹c₁₄<0, set a third row of a matrix Δ to u_3max and a fourth row to u_4min; due to¹c₁₅<0,¹c₁₆<0, set a fifth row of the matrix Δ to u_5min and a sixth row to u_6min; due to ¹c₁₇<0,¹c₁₈<0, set a seventh row of the matrix Δ to u_7min and an eighth row to u_8min; and represent an obtained matrix by Δ₁, where a boundary surface represented by the matrix Δ₁, is the key boundary surface and placed into the set Ω, and then a step 4-3-4) is performed.

${\Delta }_1=\left[\begin{array}{ccc}16& -16& -16\\ -5& -5& 5\\ 16& 16& 16\\ -5& -5& -5\\ -19.3& -19.3& -19.3\\ -9.21& -9.21& -9.21\\ -19.3& -19.3& -19.3\\ -9.21& -9.21& -9.21\end{array}\right]$

4-3-4) Due to, and ¹c₁₃·¹c₁₄<0, and ¹c₁₃>0 and ¹c₁₄<0, set the third row of the matrix Δ to theu_3min and the fourth row to the u_4max; due to ¹c₁₅<0, ¹c₁₆<0 and ¹c₁₅·u_5max+¹c₁₆·u_6min<¹c₁₅·u_5min+¹c₁₆·u_6max, set the fifth row of the matrix Δ to the u_5max and the sixth row to the u_6min; due to ¹c₁₇<0,¹c₁₈<0 and ¹c₁₇·u_7max+¹c₁₈·u_8min<¹c₁₇·u_7min+¹c₁₈·u_8max, set the seventh row of the matrix Δ to the u_7max and the eighth row to the u_8min; and represent an obtained matrix by Δ₂, where a boundary surface represented by the matrix Δ₂ is the key boundary surface and placed into the set Ω, and then a step 4-6) is performed.

${\Delta }_2=\left[\begin{array}{ccc}16& -16& -16\\ -5& -5& 5\\ -16& -16& -16\\ 5& 5& 5\\ 19.3& 19.3& 19.3\\ -9.21& -9.21& -9.21\\ 19.3& 19.3& 19.3\\ -9.21& -9.21& -9.21\end{array}\right]$

4-1) Denote the 3-5 group as a current p-q group.

4-2) Perform a step 4-4) because a third control component and a fifth control component are not a pair of linear constraint components, and m=8, which is an even number.

4-4)

4-4-1) The p-q group has three types of rectangular boundary surfaces. A step 4-4-2) is performed.

4-4-2) Construct a coordinate rotation transformation G, such that a transformed coordinate axis v₁ is perpendicular to an image of the type-III rectangular boundary surface of the group, where only a first row of the G is required because only the v₁ axis is considered.

Given that

$$\begin{gathered} B=\left(\begin{array}{cccccccc}0.9397& -0.342& 0.9563& -0.2924& 0.9659& 0.2588& 0.9744& 0.2249\\ -0.342& 0.9397& -0.2924& 0.9563& 0.2588& 0.9659& 0.2249& 0.9744\\ 1.0762& 0.179& 1.5435& -1.0787& -1.3403& -0.9867& -1.0118& 0.4121\end{array}\right),\text{ \\ G=[g11g1⁢2g1⁢3g2⁢1g2⁢2g2⁢3g3⁢1g3⁢2g3⁢3]⁢and⁢⁢E=[e1⁢1…e1⁢3e1⁢4e1⁢5e1⁢6…e1⁢8e2⁢1…e2⁢3e2⁢4e2⁢5e2⁢6…e2⁢8e3⁢1…e3⁢3e3⁢4e3⁢5e3⁢6…e3⁢8]} \end{gathered}$$

are set. E=G·B is set, such that

$\begin{array}{c}{e}_{14}=\frac{u_{3\max }-u_{3\min }}{u_{4\max }-u_{4\min }}{e}_{13},\\ e_{16}=\frac{u_{5\max }-u_{5\min }}{u_{6\max }-u_{6\min }}{e}_{15}.\end{array}$

The G and the B are substituted into the E=G·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{g}_{1s}·b_{s3}=e_{13}\\ \sum _{s=1}^3{g}_{1s}·b_{s4}=\frac{u_{3\max }-u_{3\min }}{u_{4\max }-u_{4\min }}{e}_{13}\\ \sum _{s=1}^3{g}_{1s}·b_{s5}=e_{15}\\ \sum _{s=1}^3{g}_{1s}·b_{s6}=\frac{u_{5\max }-u_{5\min }}{u_{6\max }-u_{6\min }}{e}_{15}\end{array}& \left(15\right)\end{array}$

The linear equation set (15) is solved to obtain the first row of the G, and a first row (e₁₁, . . . , e₁₈)=(0.2621 1.8566 0.5041 1.6130 1.0 2.0955 1.0152 2.4431) of the matrix E according to the E=G·B. Due to e₁₄>0 and e₁₆>0, a step 4-4-3) is performed.

4-4-3) Process other rows than third, fourth, fifth, and sixth rows of a matrix T according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and then perform a step 4-4-5).

Due to e₁₁>0 and e₁₂>0, and e₁₁·u_1max+e₁₂·u_2min<e₁₁·u_1min+e₁₂·u_2max, a first row of the matrix T is set to u_1min, and a second row is set to u_2max; due to e₁₇>0 and e₁₈>0, and e₁₇·u_7max+e₁₈·u_18min<e₁₇·u_7min+e₁₈·u_8max, a seventh row of the matrix T is set to u_7min, and an eighth row is set to u_8max; and an obtained matrix is represented by T₁, and a boundary surface represented by the matrix T₁ is the key boundary surface and placed in the set Ω

$T_1=\left[\begin{array}{cccc}-16& -16& -16& -16\\ 5& 5& 5& 5\\ 16& -16& -16& 16\\ -5& 5& 5& -5\\ -19.3& -19.3& 19.3& 19.3\\ 9.21& 9.21& -9.21& -9.21\\ -19.3& -19.3& -19.3& -19.3\\ 9.21& 9.21& 9.21& 9.21\end{array}\right]$

4-4-5) Construct a coordinate rotation transformation ²T, such that a transformed coordinate axis v₁ is perpendicular to an image of the type-I rectangular boundary surface of the group, where only a first row of the ²T is required because only the v₁ axis is considered.

Given that

$$\begin{gathered} B=\left(\begin{array}{cccccccc}0.9397& -0.342& 0.9563& -0.2924& 0.9659& 0.2588& 0.9744& 0.2249\\ -0.342& 0.9397& -0.2924& 0.9563& 0.2588& 0.9659& 0.2249& 0.9744\\ 1.0762& 0.179& 1.5435& -1.0787& -1.3403& -0.9867& -1.0118& 0.4121\end{array}\right),\text{ \\ 2T=[2t1⁢12t1⁢22t1⁢32t2⁢12t2⁢22t2⁢32t3⁢12t3⁢22t3⁢3],and⁢2⁢C=[2c1⁢1…2c13…2c15…2c182c21…2c23…2c25…2c282c31…2c33…2c35…2c38]} \end{gathered}$$

are set. ²C=²T·B is set, such that ²c₁₃=0, and ²c₁₈=0. The ²T and the B are substituted into the ²C=²T·B to obtain a following equation set:

$\begin{array}{cc}\{\begin{array}{c}\sum _{s=1}^3{2}^{}{t}_{1s}·b_{s3}=0\\ \sum _{s=1}^3{2}^{}{t}_{1s}·b_{s5}=0\end{array}& \left(16\right)\end{array}$

The linear equation set (16) is solved. Set ²t₁₃=1, such that ²t₁₃=−0.0143, ²t₁₂=5.2321 can be calculated.

Then, a first row of the matrix ²C can be calculated according to the ²C=²T·B, which is (²c₁₁, . . . , ²c₁₈)=(−0.7266,5.1005,0,3.9289,0, 4.0633,0.151,5.507). Due to ²c₁₄>0 and ²c₁₆>0, a step 4-4-7) is performed.

4-4-7) Process other rows than third, fourth, fifth, and sixth rows of a matrix Λ according to a following rule, where a boundary surface represented by an obtained matrix is the key boundary surface and placed into the set Ω; and because two key boundary surfaces have been found for the group, a step 4-6) is performed.

Due to ²c₁₁·²c₁₂<0, and ²c₁₁<0 and ²c₁₂>0, a first row of the matrix Λ is set to u_1max, and a second row is set to u_2min; due to ²c₁₇>0, ²c₁₈>0, a seventh row of the matrix Λ is set to u_7min, and an eighth row is set to u_8min; and an obtained matrix is represented by Λ₁, and a boundary surface represented by the matrix Λ₁ is the key boundary surface and placed into the set Ω.

${\Lambda }_1=\left[\begin{array}{cccc}16& 16& 16& 16\\ -5& -5& -5& -5\\ 16& -16& -16& 16\\ -5& -5& -5& -5\\ -19.3& -19.3& 19.3& 19.3\\ -9.21& -9.21& -9.21& -9.21\\ -19.3& -19.3& -19.3& -19.3\\ -9.21& -9.21& -9.21& -9.21\end{array}\right]$

4-6) Determine the key boundary surface for each group according to the steps 4-1 to 4-5-7 in this embodiment of the present disclosure.

Due to limited space, the key boundary surface is not determined for each group. Therefore, the above steps do not provide all elements of the set Ω. As a result, a step 6) is performed, while a step 5) is omitted.

Step 6) Constitute the boundary ∂(Φ_ELR) of the control attainable set by using boundary surfaces that are of the control attainable set and determined based on all key boundary surfaces in the Ω′. According to the implementation steps of the present disclosure, a total of 36 boundary surfaces of the control attainable set in this embodiment can be calculated. Because the step 5) is omitted, all the boundary surfaces of the control attainable set are not calculated any longer. Only boundary surfaces that are of the control attainable set and for the groups 1-2 and 3-5 are calculated below.

Three vertices of a key boundary surface Δ₁ in the 1-2 group are mapped according to v=B·u_ELR to obtain three vertices of a corresponding boundary surface of the control attainable set. In this way, it is determined that one boundary surface of the control attainable set is a triangle represented by a matrix ∂(Δ₁), with each column representing one vertex.

$\partial \left({\Delta }_1\right)=\left[\begin{array}{ccc}-8.3947& -38.4651& -41.8851\\ -46.8360& -35.8920& -26.4950\\ 97.1013& 62.6629& 64.4529\end{array}\right]$

Three vertices of another key boundary surface Δ₂ in the 1-2 group are mapped according to the v=B·u_ELR to obtain three vertices of a corresponding boundary surface of the control attainable set. In this way, it is determined that one boundary surface of the control attainable set is a triangle represented by a matrix ∂(Δ₂), with each column representing one vertex.

$\partial \left({\Delta }_2\right)=\left[\begin{array}{ccc}32.9753& 2.9049& -0.5151\\ -9.2454& 1.6986& 11.0956\\ -53.8688& -88.3072& -86.5172\end{array}\right]$

Four vertices of a key boundary surface T₁ in the 3-5 group are mapped according to the v=B·u_ELR to obtain four vertices of a corresponding boundary surface of the control attainable set. In this way, it is determined that one boundary surface of the control attainable set is a quadrilateral represented by a matrix ∂(T₁), with each column representing one vertex.

$\partial \left(T_1\right)=\left[\begin{array}{cccc}-32.9753& -66.5009& -33.9843& -0.4587\\ 9.2454& 28.1652& 20.3630& 1.4432\\ 53.8688& -6.3102& -39.8708& 20.3082\end{array}\right]$

Four vertices of another key boundary surface Λ₁ in the 3-5 group are mapped according to the v=B·u_ELR to obtain four vertices of a corresponding boundary surface of the control attainable set. In this way, it is determined that one boundary surface of the control attainable set is a quadrilateral represented by a matrix ∂(Λ₁), with each column representing one vertex.

$\partial \left({\Lambda }_1\right)=\left[\begin{array}{cccc}-8.3947& -38.9963& -1.7125& 28.8891\\ -46.8360& -37.4792& -27.4895& -36.8463\\ 97.1013& 47.7093& -4.0263& 45.3657\end{array}\right]$

The foregoing are merely descriptions of preferred embodiments of the present disclosure, and are not intended to limit the protection scope of the present disclosure. Any modification, equivalent substitution, improvement, etc. within the spirit and principles of the present disclosure shall fall within the scope of protection of the present disclosure.

Claims

1. A method for calculating a control attainable set of a redundant drive system under a linear constraint, comprising following steps: 1) establishing an expression of a control attainable set of a redundant drive system under each pair of linear constraint control components as follows: