Local performance optimization for a class of redundant eight-degree-of-freedom manipulators

Local performance optimization for joint limit avoidance and manipulability maximization is presented using the Jacobian matrix pseudoinverse and objective function gradient projection into the Jacobian null space. Real-time performance optimization is achieved for a member of the class of 8-joint redundant manipulators having a 3-axis spherical shoulder, a single elbow joint, and a 4-axis spherical wrist. Symbolic solutions are used for both full and partitioned Jacobian matrix pseudoinverses and objective function gradients. Results are presented to demonstrate the effectiveness of the performance optimization. A kinematic limitation of this class of manipulators and its effect on redundancy resolution is discussed.<<ETX>>

The first term of equation (2) (3) and (4a) are used for the particular solution to equation (1). If the geometric relationship of equation (3) is not exploited for the solution of 04, the pseudoinverse of the 6 x 8 Jaeobian matrix and equation (3) always yield the same value.
The hoinogeneous solution of equation (1) for local redundancy optimization is The null-space projection operator is a square matrix of order 7 that was obtained from the reduced ,]acobian matrix.  (1). In general, J J* = Ira, trot J*J _ I_. However, because of tile geometric elbow constraint of the eight-axis arm, the fourth row and cohmm of J*J are tlle same as tile fourth row and column of the identity matrix, when J is the fifll 6 x 8 ,Iaeobian matrix.
The null-space projection matrix thus has zeros for the fourth row and cohmm. This condition has two consequences.
(1) Because the fourth row is zero, a homogeneous term for 04 does not exist.
(2) Because the fourth eohmm is zero, the partial derivative of a constraint filnction with respect to 04 never adds to the homogeneous terms for the other joint rates.
The total solution to equation (1) is as follows.
The particular solution for the elbow joint rate is given in equation (3), and the total solution for the remaining joints is the sum of the particular and homogeneous solutions: where H A is a function of the arm joint angles: The total arm joint rate solution is ms follows. The particular solution for the elbow joint rate is given !n equation (3)• Again, the homogeneous solution for 04 is zero. The total solution for the remaining arm joints is the sum of the particular and homogeneous solutions:   4(b).) The zeros of DUL were obtained numerically because of its complexity.
The singularity condition for the wrist joints is independent of 05 and 0s and is given as follows: where where Or is a single constant manipulator configuration that is good for avoiding collisions with an obstacle, and W is a diagonal matrix with positive gains. Except for normalization, equation is similar to equation    Tin]e, sec (b) 0._,.   With no optinfization (k = 0), the objective fimction is greater, which means the joints are generally farther from their center of travel and thus nearer to limits. Optimization (k = -0.5) improves this situation and forces tile joints to be farther from their limits.

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A compelling demonstration of the benefit of joint limit avoidance optimization shows when a joint limit is encountere(t. For the stone trajectory, figure 6(b) shows 05 hitting a linfit at 9.5 sec, t)ut avoiding the limit when tile function is minimized. The associated Cartesian error due to the joint limit is shown ill figure 6(c     fig. 10(a) Figure  12 shows joint limit avoidance for the full (k =-0.5, repeated from fig. 6(a)) and the partitioned (k A = kw = -0.5) solutions.
The     I I I I t I I  I  5  10  15  20  25  30  Time, sec Full vs partitioned manipulability maximization.
12 Figure  13 shows that the wrist manipulability for the partitioned case increases rapidly to v_ and is held there for the remaining trajectory.
The partitioned arm manipulability increa_ses to a lower value and falls off as the elbow work space limit singularity is approached. The full manipulability increases to a value in between the arm and wrist curves. It also falls off as the elbow work space limit singularity is approached. Therefore, the wrist nmnipulability appears to be superior to the flfll solution, and the arm manipulability tends to be lower than the full solution. simultaneously, and the full and partitioned IJJTI were studied. Figure  14 shows the manipulator in these singular conditions, where the full configuration is 0 = {0, 0, 90, 70, 0, 90, -90, 0} :r. This configuration is a combination of those shown in figures 4(a) and 4(c). Figure 15 presents tile results of this study. Starting from 0= {0, -10, 80, -70, 0, 80, -100, 0} T, .ioints 2, 3, 6, and 7 were updated by l°/sec, so both singularities were reached at 10 sec. Figure  15 is a plot of IJsxvJ_'x71 (5 x 7), lJlu,r.4JT,_r.4l (arm) and IJLRJTRI (wrist). Tile arm curve is symnmtric; the 5 x 7 and wrist curves arc not because joint 6 hits a limit at the wrist singularity and does not. move through.

Eight-Axis Arm Design Limitation
This section discusses a limitation in the eightaxis arm design regarding redundancy optimization. Design alternatives are presented in appendix D to alleviate the problem.
As discussed in section 3.2.1, the length of reach from shoulder to wrist for manipulators with a spher-ical wrist, spherical shoulder, and a single elbow joint is a function of only the elbow joint angle. Figure  3 shows this relationship. The Jacobianmatrix for the eight-axisarm expressed in theelbowCartesian coordinate frame,{4}, is presented in reference 13asfollows: Therearetrade-offs amongtile originaldesign of figure1andtheproposed redesigns offigures D1,D2, andD3. Theoriginal designhassimplerkinematics, joint 4 variablessolvedindependently, and allows an efficientpartitionedsolution. However, joint 4 hasnonull space andonlyonemodeof self-motion.
The eight-axis redesign concepts providetwo modes of self-motion, a null-space termassociated with 0,1, Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate