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UNC Chapel Hill M. C. Lin COMP Robotics: An Introduction Kinematics & Inverse Kinematics

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UNC Chapel Hill M. C. Lin Forward Kinematics

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UNC Chapel Hill M. C. Lin What is f ?

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UNC Chapel Hill M. C. Lin What is f ?

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UNC Chapel Hill M. C. Lin Other Representations Separate Rotation + Translation: T(x) = R(x) + d –Rotation as a 3x3 matrix –Rotation as quaternion –Rotation as Euler Angles Homogeneous TXF: T=H(R,d)

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UNC Chapel Hill M. C. Lin Forward Kinematics As DoF increases, there are more transformation to control and thus become more complicated to control the motion. Motion capture can simplify the process for well-defined motions and pre-determined tasks.

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UNC Chapel Hill M. C. Lin Forward vs. Inverse Kinematics

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UNC Chapel Hill M. C. Lin Inverse Kinematics (IK) As DoF increases, the solution to the problem may become undefined and the system is said to be redundant. By adding more constraints reduces the dimensions of the solution. It’s simple to use, when it works. But, it gives less control. Some common problems: –Existence of solutions –Multiple solutions –Methods used

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UNC Chapel Hill M. C. Lin Numerical Methods for IK Analytical solutions not usually possible –Large solution space (redundancy) –Empty solution space (unreachable goal) f is nonlinear due to sin ’s and cos ’s in the rotations. –Find linear approximation to f -1 Numerical solutions necessary –Fast –Reasonably accurate –Yet Robust

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UNC Chapel Hill M. C. Lin The Jacobian

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UNC Chapel Hill M. C. Lin The Jacobian

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UNC Chapel Hill M. C. Lin The Jacobian

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UNC Chapel Hill M. C. Lin Computing the Jacobian To compute the Jacobian, we must compute the derivatives of the forward kinematics equation The forward kinematics is composed of some matrices or quaternions

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UNC Chapel Hill M. C. Lin Matrix Derivatives

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UNC Chapel Hill M. C. Lin Rotation Matrix Derivatives

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UNC Chapel Hill M. C. Lin Angular Velocity Matrix

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UNC Chapel Hill M. C. Lin

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UNC Chapel Hill M. C. Lin

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UNC Chapel Hill M. C. Lin Computing J + Fairly slow to compute –Breville’s method: J + (JJ T ) -1 Complexity: O(m 2 n) ~ 57 multiply per DOF with m = 6 Instability around singularities –Jacobian loses rank in certain configur.

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UNC Chapel Hill M. C. Lin Jacobian Transpose Use J T rather than J + –Avoid excessive inversion –Avoid singularity problem

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UNC Chapel Hill M. C. Lin Principles of Virtual Work Work = force x distance Work = torque x angle

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UNC Chapel Hill M. C. Lin Jacobian Transpose Essentially we’re taking the distance to the goal to be a force pulling the end-effector. With J -1, the solution was exact to the linearized problem, but this is no longer so.

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UNC Chapel Hill M. C. Lin Jacobian Transpose

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UNC Chapel Hill M. C. Lin Jacobian Transpose In effect this J T method solves the IK problem by setting up a dynamical system that obeys the Aristotilean laws of physics: F = m v ; = I and the steepest descent method. The J + method is equivalent to solving by Newtonian method

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UNC Chapel Hill M. C. Lin Pros & Cons of Using J T + Cheaper evaluation + No singularities - Scaling Problems -J + has minimal norm at every step and J T doesn’t have this property. Thus joint far from end-effector experience larger torque, thereby taking disproportionately large time steps -Use a constant matrix to counteract - Slower Convergence than J + -Roughly 2x slower [Das, Slotine & Sheridan]

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UNC Chapel Hill M. C. Lin Cyclic Coordinate Descend (CCD) Just solve 1-DOF IK-problem repeatedly up the chain 1-DOF problems are simple & have analytical solutions

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UNC Chapel Hill M. C. Lin CCD Math - Prismatic

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UNC Chapel Hill M. C. Lin CCD Math - Revolute

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UNC Chapel Hill M. C. Lin CCD Math - Revolute You can optimize orientation too, but need to derive orientation error and minimize the combination of two You can derive expression to minimize other goals too. Shown here is for point goals, but you can define the goal to be a line or plane.

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UNC Chapel Hill M. C. Lin Pros and Cons of CCD + Simple to implement + Often effective + Stable around singular configuration + Computationally cheap + Can combine with other more accurate optimizations - Can lead to odd solutions if per step not limited, making method slower - Doesn’t necessarily lead to smooth motion

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UNC Chapel Hill M. C. Lin References

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