Fig 1.
Geometry of the Jacobian matrix.
The ith column of the Jacobian is formed using the joint axis and translation vector to the endpoint ri.
Fig 2.
The Cartesian stiffness ellipsoid of a serial link mechanism.
As the mechanism approaches a singular configuration the longest axis of the Cartesian stiffness ellipsoid grows to infinity.
Fig 3.
Optimal stiffness for revolute joints.
In this configuration the forces and moments on the endpoint produce zero joint torque.
Fig 4.
Alternate optimal stiffness for a revolute joint.
In this configuration the endpoint forces and moments negate each other.
Fig 5.
Optimal stiffness for a prismatic joint.
Applied forces must be orthogonal to the axis of actuation.
Fig 6.
Optimal compliance in revolute joints.
The moments from the endpoint forces are parallel to the joint axes, maximising joint torque and hence joint deflection.
Fig 7.
Optimal compliance in a prismatic joint.
The applied forces are orthogonal to the axis of actuation.
Fig 8.
A singular configuration in the human arm.
An outstretched arm leads to a kinematic singularity in the direction from the shoulder to the hand. This results in maximum stiffness.
Fig 9.
Stiffness ellipsoid in a human arm.
The principle radii correspond to directions of maximum stiffness, and maximum compliance respectively.
Fig 10.
Stiffness is relative to the direction of applied forces.
Forces in region (a) lead to shoulder muscle activation, forces in region (b) lead to elbow muscle activation, and forces in region (c) create both.
Fig 11.
The 8 inverse kinematics solutions for the UR3 robot arm.
The relative stiffness in the x-direction is compute for each configuration using Eq 13.
Fig 12.
Rotoscoped images of the Cartesian stiffness ellipsoid with changing arm configuration.
A robot can be made to increase stiffness or compliance in different directions. Each case begins from the same starting configuration.
Fig 13.
Comparison of a humanoid robot to human experiments.
Applying the optimisation principle to a humanoid robot (left) results in similar arm configurations as observed in human.
Fig 14.
A serial link mechanism becomes more compliant when pushed in one direction (left), and more stiff when pulled in the other (right).