DRAWINGS—FIGURES
FIG. 1 is a perspective view of the manipulator arm with all links contracted.
FIG. 2 shows an end view of the links with a payload attached to the last link.
FIG. 3 illustrates the deployment scheme for the arm.
FIG. 4 is a schematic of a 2-link arm for dynamic modeling.
FIG. 5 is a perspective view of the preferred embodiment of a 3-link arm.
FIG. 6 shows the variation of the configuration dependent modulating coefficients.
FIG. 7 shows a typical polynomial sigmoid trajectory with all the parameters.
FIG. 8 shows the overall control scheme under disturbances.
FIG. 9 shows simulation results for the control algorithm.
FIG. 10 is an image of a 3 link robot arm with one actuated and two unactuated joints.
DETAILED DESCRIPTION
Overview
The current invention pertains to the design and control of a robot arm capable of automated assembly operations inside an aircraft wing. Most assembly operations in aircraft manufacturing are currently done manually. A worker enters the wing through small access portholes and lies flat on the base, while carrying out the assembly operations. Evidently the working conditions are ergonomically challenging. The size and weight of manipulator arms have been the primary impediments in the automation process.
We propose a deployable serial linkage structure for the manipulator arm as shown in FIG. 1. The links are aluminum C channels (50-53) with successively smaller base and leg lengths. The links are connected by 1 degree of freedom rotary joints (54-56). The use of a channel structure is advantageous for a number of reasons. The channels can fold into each other resulting in an extremely compact structure during entry through the access porthole. Once inside the wing, the links may be deployed to access a number of assembly points. The open channel structure also facilitates the attachment of a payload 57 to the last link, as shown in FIG. 2.
The deployment scheme modulates gravitational torques on the links to be deployed by using just one actuator at the base link. The deployed link is locked once it reaches the desired position. The use of a single actuator at the base drastically reduces the weight and size of the robot arm. We also propose an algorithm for point to point control of the links to be deployed.
Gravity Modulation
FIG. 3 illustrates the basic deployment process for the linkage structure shown in FIG. 1. There is no dedicated actuator at the individual joints (54-56) along the arm linkage. The only actuated link is 50 which can be rotated about axis 58. The axis of rotation 58 is orthogonal to the direction of gravity. Each joint is free to rotate unless a locking mechanism (not shown) fixes the joint.
As shown in FIG. 3(a), the first step is to free rotary joint 54 and lock rotary joints 55 and 56. Then link 50 is rotated in the counter-clockwise direction. This tends to rotate the free link 51 due to gravity. After arriving at a desired angle, 180 degrees in FIG. 3(a), rotary joint 54 is locked and joint 55 is unlocked. At this time the actuated link 50 is rotated in the clockwise direction, as shown in FIG. 3(b). This allows link 52 to rotate so that it is deployed as seen in FIG. 3(b).
This procedure can be repeated as many times as the number of arm joints. The only actuator needed for this deployment operation is the actuator for link 50 in conjunction with locking mechanisms at individual joints. Contraction of the arm can be performed by reversing the above deployment procedure. Starting with the tip joint, individual joints can be closed one by one towards the first joint.
Dynamic Modeling
FIG. 4 shows a schematic of a 2-link robot arm with the base link 50 actuated and the 2nd link 51 unactuated. The base link 50 may be rotated about axis 58 by an actuator (not shown).
The angles θ1 and θ2 are measured as shown in FIG. 4. We seek rotation of free link 51 about axis 54 (Z1) by rotating the actuated link 50 about axis 58 (Z0). It is intuitively obvious that by rotating actuated link 50 about axis 58, we can achieve a rotation of free link 51 about axis 54 because of the gravitational torque. The axis of rotation 54 of free link 51 is also rotating about axis 58. This results in an additional dynamic coupling, as seen in the analysis that follows.
This idea can be extended to multiple serial links. The gravitational and gyroscopic torques may be used to actuate the links, one at a time, by designing a suitable θ1 trajectory for the actuated link 50. All other links must be locked prior to the actuation of the target link.
The advantage of such a system is the drastic reduction in the number of actuators required to reconfigure the structure. The presence of actuators at each rotary joint would have made the system extremely bulky and unsuitable for our application. Our proposed scheme uses a single actuator and thus results in a very compact structure which is scalable to multiple links.
We analyze the system in order to determine the input-output relationship between the actuated and underactuated joints. Lagrange's equations of motion for the 2-link robot arm can be written as:
The equation of motion of the unactuated link may be written as:
Mi, Ixxi etc. denote the mass and inertias of link i. xci, ai etc. denote the distance of the center of mass and the Denavit-Hartenberg parameters of the ith link with respect to the (i-1)th coordinate system.
It may be shown that (2) is a 2nd order non-holonomic constraint and thus cannot be integrated to express θ2 as a function of θ1. It is sufficient to determine desired θ1 trajectories θ1d(t)) in order to achieve point to point control of θ2. Once θ1d(t) is obtained, we can set the input joint torque τ1 to be:
τ1=(({umlaut over (θ)}1d−2λ θ1−λ2θ1)+F1+G1)/N11 (3)
Here N=H−1 and θ1=θ1d−θ1. By choosing the gain A appropriately, we can ensure that the resulting error dynamics is exponentially stable.
We first explore the qualitative behavior of the differential equation expressing the 2nd order nonholonomic constraint in order to better understand the dominant dynamic effects. We refer to the terms involving θ1 and its derivatives as the control input and terms involving θ2 as the modulating coefficients. The modulating coefficients are solely dependent on the angular position of the unactuated link, whereas we can design the control input so as to get a desired motion of the unactuated link.
FIG. 6 shows the variation of the modulating coefficients with angular positions θ2. We note that θ2 ranges from 90° to 270° in our coordinate system. The parameter values are taken from our actual robotic system, which is shown in FIG. 10. Clearly, the dominant term is the modulating coefficient G2 due to gravity, followed by the contribution of the inertial term H12 and finally the contribution of the centrifugal term F2. We also identify points in the configuration space of the unactuated coordinate where the modulating coefficients change sign. We will use these features in the design of control inputs so as to get desired outputs for the unactuated coordinate.
Control Algorithm
There are 3 regimes of motion of the unactuated coordinate θ2 based on the sign of the dominant modulating coefficient G2:
1. G2(θ2)>0 during motion
2. G2 (θ2) <0 during motion
3. G2 (θ2) changes sign during motion
From (2), we may conclude that the control input θ1 must start from 0 and return to 0 at the end of the motion. Further, we may infer that the control input θ1 undergoes at least one change of sign when the motion of the unactuated coordinate is in the 1st or 2nd regime. In the 3rd regime, no change of sign is necessary. We construct the θ1 trajectory by smoothly patching together 3 piecewise polynomial sigmoid segments, as shown in FIG. 7.
We parameterize the θ1 trajectory as follows:
θ1(t)=[10(t/tf1)3−15(t/tf1)4+6(t/tf1)5]θ1a0≦t≦tf1
θ1(t)=[10(tf2−t/tf2−tf1)3−15(tf2−t/tf2−tf2)4+6(tf2−t/tf2−tf1)5](θ1a−θ1h)+θ1htf1≦t≦tf2 (4)
θ1(t)=[10(tf−t/tf−tf2)3−15(tf−t/tf−tf2)4+6(tf−t/tf−tf2)5]θ1htf2≦t≦tf
We need to determine the parameters θ1a, θ1b, η1, η2 and tf of the θ1 trajectory for point to point motion of θ2 between θ20 and θ2f. We do this by substituting the parameterized control input in (2) and solving it as a 2 point boundary value problem (bvp). The system (2) becomes a 2nd order bvp with 4 boundary conditions and 5 unknown parameters to be determined. The boundary conditions are:
θ2(0)=θ20, {dot over (θ)}2(0)={dot over (θ)}20, θ2(tf)=θ2f{dot over (θ)}2(tf)={dot over (θ)}2f
This system is clearly indeterminate. We thus fix 3 of the unknown parameters, viz. η1, η2 and tf, and solve the 2nd order bvp for θ1a and θ1b. This is motivated by the fact that θ1a and θ1b are linearly involved parameters if we ignore the weak term associated with {dot over (θ)}12. The parameter values η1 and η2 are fixed such that η1=η2−η1=1−η2=⅓. We note that if θ1(t) (with parameters θ1a and θ1h) is an input trajectory for motion of the unactuated coordinate from θ20 to θ2f in time tf, {dot over (θ)}1(t)=θ1(tf−t) is the input trajectory for motion from θ2f to θ20. Since η1=η2−η1=1−η2=⅓ the parameters for the sigmoid trajectory for retraction are {dot over (θ)}1a=θ1b hand {dot over (θ)}1b=θ1a. Thus, we do not need to recompute the parameters of the sigmoid trajectory for retraction of the free link 51. The parameter tf may be set to get a desired average speed of motion required for point to point movements.
In the simulation results, 3 of the parameters were fixed at η1=⅓, η2=⅔ and tf=4. It should be noted that other solutions may be obtained by changing η1 and η2, but we need to recompute the parameters θ1a and θ1h for retraction. The results are shown in FIG. 9(a) for θ2(0)=110°, {dot over (θ)}2(0)=0, θ2(tf)=150°, {dot over (θ)}2(tf)=0. The 2 unknown parameters for the θ1 trajectory are θ1a=76° and θ2a=1.04°. FIG. 9(b) shows the results for θ2(0)=130°, {dot over (θ)}2(0)=0, θ2(tf)=250°, θ2(tf)=0. The 2 unknown parameters for the θ1 trajectory are: θ1a=3.13° and θ2a=2.63°. As desired, the motion of the base link is restricted to very small amplitudes in both cases.
FIG. 8 shows the overall control scheme for a 2-link arm in the presence of disturbances. There may be disturbances acting on the unactuated joint 54 during the motion of the unactuated link 51 causing it to deviate from its predicted trajectory. The initial motion plan for the actuated joint 58 is generated by the initial trajectory generator 70. The actuated joint 58 is controlled through a local feedback loop 71. The motion plan for the actuated joint 58 is updated by the dynamic trajectory planner 73 based on actual measurements of position and velocity 72.
Embodiments
FIG. 5 shows the preferred embodiment of the robot arm with 3 C-links 50-52. A T-link 61 is rigidly connected to link 50. An AC servo motor (with optical encoder) 59 coupled to harmonic drive gearing 60 is used as a backlash free actuation mechanism. This mechanism is used to rotate the T-link 61 and link 50 about axis 58. This embodiment has optical encoders 62 at the free joints for measuring angular positions of the unactuated links 51-52. This embodiment also uses pneumatic brakes 63 as locking mechanisms at the free joints.
FIG. 10 shows an image of the preferred embodiment of the robot arm with 3 C-links 50-52. The arm is inside a mock-up of an airplane wing box 65. The arm enters the wing box 65 through an access porthole 64. There is also a payload 57 attached to the terminal link 52.
Patent Application
20080028880
