1. ROBOTICS 01PEEQW Basilio Bona DAUIN – Politecnico di Torino

2. Statics

3. Statics – 1 Generalized Forces  We call Generalized Forces the vector of forces and torques ‒It is not a vector in strict terms because the elements have different units (forces are expressed in N, torques in N·m  Statics studies the relations between the task space generalized forces (TSGF) and the joint generalized forces (JGF) in static equilibrium conditions  The TSGF are generated from interactions with the environment (e.g., when the TCP pushes against a surface)  The JGF are generated by the power supplied by the joint motors used to move the robot arms 3 ROBOTICS 01PEEQW - 2015/2016

4. Statics – 2 4 ROBOTICS 01PEEQW - 2015/2016 BASE TCP Cartesian (task space) generalized forces Joint generalized forces

5. Statics – 3  Prismatic joint torques  Revolute joint torques  To find the relation between we apply the virtual work principle  TCP generalized forces define a virtual work  Joint generalized forces define another virtual work 5 ROBOTICS 01PEEQW - 2015/2016

6. Statics – 4 static equilibrium  Virtual work principle says that a static equilibrium condition exists when  Virtual displacements are “similar” to differential displacements, i.e.,  Recalling that 6 ROBOTICS 01PEEQW - 2015/2016 This is the relation between TCP forces and joint forces. equivalence It is an equivalence relation equilibrate If one needs to compute the joint forces needed to equilibrate the TCP force, the relation is Equilibrate and Balance are synonymous

7. Kineto-static duality – 1  Since kineto-static duality we speak of a kineto-static duality between generalized (cartesian) forces and cartesian velocities. Considering the geometric Jacobian (that has is geometrically more significant than the analytical one) we have 7 ROBOTICS 01PEEQW - 2015/2016  The duality can be characterized considering the mathematical concepts of range and kernel of the transformations

8. Matrix review (from MSMS course) – 1 8 ROBOTICS 01PEEQW - 2015/2016

9. Matrix review (from MSMS course) – 2 9 ROBOTICS 01PEEQW - 2015/2016

10. Matrix review (from MSMS course) – 3 10 ROBOTICS 01PEEQW - 2015/2016

11. Kineto-static duality – 2 11 ROBOTICS 01PEEQW - 2015/2016  Image space It contains the TCP velocities that can be generated by the joint velocities, for a given pose  Null space It contains the joint velocities that do not produce any TCP velocities, for a given pose  Consider the joint torques  Image space It contains the joint generalized torques that can balance TCP generalized forces, for a given pose  Null space It contains the TCP generalized forces that do not require balancing joint generalized forces, for a given pose  Consider the Cartesian velocity

12. Kineto-static duality – 3 singular  When the robot is in a singular configuration: 12 ROBOTICS 01PEEQW - 2015/2016 There are non zero joint velocities that produce zero TCP velocities There are non zero joint generalized forces that cannot be balanced by TCP generalized forces There are TCP generalized forces that do not require any balancing joint generalized forces There are TCP velocities that cannot be obtained by any joint velocities See Example_2014_02

13. Elasticity of the structure  In real conditions a perfectly rigid robot does not exist  Elastic effects are always present and can be localized in: 1.Joints, due to the mechanical transmission elements: long motor shafts, belts, chains, gearboxes, etc. 2.Links, due to distributed compliance of the mechanical structure (flexion, torsion, compression) 13 ROBOTICS 01PEEQW - 2015/2016 21

14. Elasticity – 1  When a generalized force is applied to the robot TCP a small deflection takes place  We want to describe the relation in static conditions between the relevant variables  We introduce an approximated description, considering the elasticity due only to the joints (links are assumed perfectly rigid) 14 ROBOTICS 01PEEQW - 2015/2016

15. Elasticity – 2 15 ROBOTICS 01PEEQW - 2015/2016

16. Elasticity – 3 16 ROBOTICS 01PEEQW - 2015/2016

17. Elasticity – 4 17 ROBOTICS 01PEEQW - 2015/2016

18. Conclusions  Statics is important since it allows to compute the equivalent effects on joints of TCP forces when the TCP interacts with a surface (and viceversa) kinetostatic duality  Statics and velocity kinematics are linked by kinetostatic duality  Remember that the product of a force by a velocity is a power  For this reason forces and velocities cannot be set independently when the power is an external constraint  If you set a force you cannot set also the corresponding velocity and viceversa  Elastic forces are usually not considered in the robot model, but they are very important for the control design of real systems 18 ROBOTICS 01PEEQW - 2015/2016