Measure of Manipulability for Robot Manipulators

What they do

Choosing an arm’s mechanism and link lengths, and where and in what posture it should work, had been done by intuition and experience. Yoshikawa earlier proposed the measure of manipulability, w = √det(J Jᵀ), to quantify how easily an arm can move its end-effector’s position and orientation. This paper derives further properties of w and finds the posture maximizing it, the “optimal posture”, for common arm types and a robot finger.

How it works

• Ellipsoid view. w is the product of J’s singular values and is proportional to the volume of the manipulability ellipsoid: the end-effector velocities reachable with joint velocity norm at most 1. For a non-redundant arm, w =

  det J

  .

• New properties. For a non-redundant arm, joint speeds bounded by

  θ̇i

  ≤ 1 map to a parallelepiped of volume 2ᵐw. Unequal joint speed limits and task speeds are handled by rescaling J, which for a non-redundant arm only multiplies w by a constant.

• Kinematics only. Unlike Asada’s generalized inertia ellipsoid, w ignores dynamics.
• Arm studies. Closed-form w for two-link, SCARA, PUMA (main three joints, hand position) and Cartesian, cylindrical and polar arms; numerical maximization for a 4-joint finger.

Results

• Two-link and SCARA arms: w = l1 l2

  sin θ2

  , best at θ2 = ±90° for any l1, l2, θ1, and with l1 + l2 fixed, best at l1 = l2. A human arm roughly has l1 = l2 and bends the elbow about 90° when handling objects with both hands.

• PUMA: best with the hand level with the second joint and θ3 slightly below 90°, which moves the hand farther from joint 1’s axis.
• Cartesian arms have w = 1 everywhere; cylindrical and polar arms are best fully extended, at the workspace boundary.
• 4-joint finger: w peaks at fingertip distance 0.8 for planar motion and 0.9 with all four joints, postures the author finds close to a human pinching a small object. Salisbury and Craig’s condition number gives different optima.

Limitations

Analysis and numerical computation only, with no robot experiments; the PUMA wrist is ignored. For cylindrical and polar arms the measure alone points to the workspace boundary, so other criteria must set the posture. Using w for arm design and task planning is left to future work.