A robust, optimal, adaptive technique for compensating rate and position limits in the joints of a six-degree-of-freedom manipulator has been developed. In this new algorithm, the unmet demand as a result of actuator saturation is redistributed among the remaining unsaturated joints. The scheme is used to compensate for inadequate path planning, problems such as joint limiting, joint freezing, or even obstacle avoidance, where a desired position and orientation are not attainable due to an unrealizable joint command. Once a joint encounters a limit, supplemental commands are sent to other joints to best track, according to a selected criterion, the desired trajectory.

A standard six-degree-of-freedom manipulator has six independently controlled joints. The position and orientation of the end effector, each of which is described in three dimensions, are fully determined by the angles of the joints. As long as the appropriate joint angles are achievable, the desired position and orientation can be obtained. However, when the specified joint trajectories cannot be followed due to a command beyond the range of the actuator, positions and orientations downstream from the limited joint will all be affected, causing in some cases extreme deviations from the expected values. This new scheme is an ideal solution candidate for this problem. It was designed to compensate for actuator saturation in a multivariable system by supplementing the commands to the remaining actuators to produce the desired effect on the output, in this case the gripper position and orientation. For each joint which saturates, a degree of freedom is lost, but the remaining joints can be used to track the desired path within the physical limits of the manipulator.

The matrix known as the Jacobian, *J*, describes how a small change in the joint positions, *dq*, affects the gripper. The resulting position and orientation change of the end effector, *D*, is computed as *D = Jdq*. When a joint is commanded to move beyond its limit, a portion of the command cannot be achieved. This unmet demand, * Δq*, represents the amount the joints should move but cannot. The resulting error in position and orientation of the end effector can be approximated by *D = JΔq*. The objective of this new scheme is to duplicate *D* as closely as possible using joints with authority remaining.

The figure shows the scheme with the optimal gains in the feedback loop. The commands, *q*, are checked to verify that they will not drive any of the joints to a rate or position limit. Any portion of a command which would cause a joint to saturate corresponds to unmet demand and is truncated and redirected to the feedback gains. The gains take this unmet demand, * Δq*, and produce some supplemental commands to unsaturated joints, *q**, such that *JΔq *and* Jq** are as close as possible. These supplemental commands allow the end effector to optimally track its desired trajectory, even in the face of joint position and rate limits. Since the algorithm acts upon the joint commands only, there is never the possibility of an unstable system resulting from the use of this algorithm.

The optimal feedback gains are computed using a quadratic objective function with task-dependent weights assigned to the components of the position and orientation vector of the end effector. The gains adapt to changes in the Jacobian as the manipulator moves through its workspace, and the computations are robust to singularities arising from particular manipulator configurations. This provides smooth, continuous variation of the optimal gains for as long as * Δq* is nonzero.

*This work was done by Ten-Huei Guo of *Lewis Research Center,*Jonathan Litt of the Vehicle Technology Center of the U.S. Army Research Laboratory, and André Hickman of Morehouse College.*

*Inquiries concerning rights for the commercial use of this invention should be addressed to*

###### NASA Lewis Research Center

Commercial Technology Office

Attn: Tech Brief Patent Status

Mail Stop 7 - 3

21000 Brookpark Road

Cleveland

Ohio 44135

*Refer to LEW-16566.*

##### Motion Control Tech Briefs Magazine

This article first appeared in the June, 1998 issue of *Motion Control Tech Briefs* Magazine.

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