Basilio Bona DAUIN – Politecnico di Torino 07/08/2018 ROBOTICS 01PEEQW Basilio Bona DAUIN – Politecnico di Torino di 23

Planar 2 DOF manipulator Kinematics

Planar two-arm manipulator – 1 This simple manipulator cannot provide 3 DOF, since it has only two joints This is the TCP orientation assumed as the third Euler angle We will compute below the kinematics functions of the reduced DOFs Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Planar two-arm manipulator – 2 Direct position KF, assuming Direct velocity KF Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Planar two-arm manipulator – 3 Analytical Jacobian Geometric Jacobian, assuming All joints are revolute, hence: Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Planar two-arm manipulator – 4 Geometric Jacobian Remember: all vectors are represented in Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Planar two-arm manipulator – 5 Geometric Jacobian Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Planar two-arm manipulator – 6 Geometric Jacobian These rows are canceled since they add no information In conclusion Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Inverse position – 1 Inverse position; the solution can be obtained in different ways; this is one of them: hence: To avoid inverting cos(.), we can find the solution computing Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Inverse position – 2 hence The square root produces two solutions: elbow up and elbow down elbow up elbow down Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Inverse position – 3 Angle is obtained computing Hence, solving the system Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Inverse position – 4 An alternative solution Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Inverse velocity – 1 First we reduce the Jacobian to a square matrix, deleting the angle-related row The determinant is The inverse is Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Singularity – 1 Singularity arises when Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Singularity – 2 SINGULARITY These two vectors span a subspace of dimension 1 NON SINGULARITY These two vectors span a subspace of dimension 2 Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Singularity – 3 Now we compute the various subspaces when Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Singularity – 4 Now we compute the various subspaces when Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Singularity – 5 Now we compute the various subspaces when Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Singularity – 6 Numerical example 3 1 2 4 1 2 Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Geometry of singularity – 1 4 1 Basilio Bona ROBOTICS 01PEEQW - 2015/2016

Geometry of singularity – 2 3 Basilio Bona ROBOTICS 01PEEQW - 2015/2016