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CHAPTER 6

CHAPTER 6. Robot Compliance & Force Control. x. f dist. k e. m. f. B e. Pure Force Control Along a Single Degree-of-Freedom. (Ref: Craig’s Book, Chapt. 11). Control of a Mass-Spring damper system.

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CHAPTER 6

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  1. CHAPTER 6 Robot Compliance & Force Control

  2. x fdist ke m f Be Pure Force Control Along a Single Degree-of-Freedom (Ref: Craig’s Book, Chapt. 11) Control of a Mass-Spring damper system We model the constant with the environment as a mass-spring-damper system. Environment has a mass m, stiffness ke, and damping Be.

  3. Pure Force Control Along a Single Degree-of-Freedom (Ref: Craig’s Book, Chapt. 11) • m= consists of everything beyond the wrist force sensor, • (e.g. inertia of end-effector + tool) • ke= includes compliance of surface contacted, passive compliance of wrist & force sensor. • fdist= unknown disturbance, • (e.g. friction, cogging in manipulator’s gearing)

  4. Or written in terms of the variable we wish to control, fe, (using ), Pure Force Control Along a Single Degree-of-Freedom (Ref: Craig’s Book, Chapt. 11) The variable we wish to control is the force acting on the environment fe, (w/c is the force acting in the spring), (1) Governing Eq is (2) (3)

  5. Pure Force Control Along a Single Degree-of-Freedom (Ref: Craig’s Book, Chapt. 11) Like inverse dynamics control, (computed torque), we apply a control law (4) Where ef = fd - fefd = desired force Resulting in the following closed loop system (5)

  6. (7) Pure Force Control Along a Single Degree-of-Freedom (Ref: Craig’s Book, Chapt. 11) But, we cannot use knowledge of fdist in our control law (3), therefore (3) is not feasible. Say, use (as control law) (6) Leave out fdist Equating (6) and (3) yields,

  7. Pure Force Control Along a Single Degree-of-Freedom (Ref: Craig’s Book, Chapt. 11) A Steady State analysis of (7), by setting all time derivatives to zero, results in (8) where If keis large (stiff),  is small, ef is large.

  8. Pure Force Control Along a Single Degree-of-Freedom (Ref: Craig’s Book, Chapt. 11) A better approach is to use fd in control law (4) instead of fe + fdist: (9) Equating (9) & (3) yields & a Steady State analysis (time-der= 0) yields the ff steady state error: But ef = fd - fe (9) (9) is quite an improvement over (8).

  9. Force Trajectories are usually constants. • Applications that require constant forces to follow a specified trajectory in time are rare. • Sensed forces are quite noisy, numerical differentiation to compute is ill-advised. However, , manipulators usually have sensors for . Practical Considerations

  10. Practical Considerations This leads to ff control law (10)

  11. Practical Considerations • Force errors generate a set-point for an inner velocity control loop with gain kvf. • Some force control laws also include an integral term to improve steady-state performance. • One important remaining problem is that ke appears in our control law, but is often unknown. • assume ke range, and gains are chosen such that the system is somewhat robust with respect to the variations in k.

  12. HYBRID POSITION/FORCE CONTROL Ref: M. Raubert & J. Craig, “Hybrid Position/Force Control of Manipulators”, Journal of Dyn. Syc. Meas. & Control, June ,1981. • Concept of a “Constraint Frame” on “Compliance Frame” & Framework for Partially Constrained Tasks. • Every manipulation task can be broken down into subtasks that are defined by a particular contact situation occurring between manipulator tool (n e-e) and the environment. • With each subtask, we may associate a set of constraints, called natural constraints that result from the particular mechanical and geometric characteristics of the task configuration.

  13. HYBRID POSITION/FORCE CONTROL • In general, for each subtask configuration a generalized surface can be defined with (natural) position constraints normal to the surface and (natural free constraints) along the tangents. These 2 types of constraints partition the degrees-of-freedom of possible end-effector motions into 2 orthogonal sets that must be controlled according to different criteria. • Additional constraints, called artificial constraints, are introduced in accordance with natural constraints to specify desired motions & force trajectories. • That is, each time a user specifies a desired trajectory in either position or force, an artificial constraint is defined.

  14. HYBRID POSITION/FORCE CONTROL • These artificial constraints also occur along the tangents and normals of the generalized constraint surface; but while nat’l constraints, artificial force constraints are specified along surface normals, and artificial positions constraints along tangents. Hence consistency with natural constraints is preserved.

  15. HYBRID POSITION/FORCE CONTROL • Extreme Cases: • Left figure:manipulator is moving through free space • Natural constraints are all force constraints. (all forces are zero) • Artificial constraints are all position, (position trajectory to be controlled) • Right figure:manipulator is glued to wall, cannot move • Natural constraints are all position constraints. (position = constant, vel=0) • Artificial constraints are all force, (force trajectory to be controlled)

  16. HYBRID POSITION/FORCE CONTROL • Hybrid Position/Force Controller must solve 3 problems: • Position Control along directions in which a natural force constraint exists. • Force Control of a manipulator along direction in which a natural position constraint exists. • A scheme to implement the arbitrary mixing of these modes along orthogonal degrees-of-freedom of an arbitrary frame (constraint frame)(via a Selection Matrix S). S = Selection Matrix (diagonal matrix where a unity diagonal element represents position control in that direction, 0force control). Expressed in constraint frame

  17. HYBRID POSITION/FORCE CONTROL It is better to implement Hybrid Position/Free Control using Inverse Dynamics in Cartesian (Task, or Constraint) space. Cartesian space Joint space

  18. STIFFNESS & COMPLIANCE • Force control is difficult to accomplish with rigid structures. • One way to alleviate is with the use of passive compliance devices. Devices composed of springs & dampers for the purpose of reducing the endpoint stiffness. With such a device, certain applications can be achieved with pure position control. Or better still, build compliance in the robot structure itself!

  19. STIFFNESS & COMPLIANCE • The disadvantage of passively compliant device is that they are limited in their range of applicability. • The RCC device, for example, can only insert pegs of a certain length & orientation with respect to the hand. • To achieve a wider applicability, active control of end-point compliance is necessary.

  20. STIFFNESS CONTROL OF A SINGLE Degree-of-freedom Ref: Spoy & Vidyasajar’s work, Ch 9. Environment Assume manipulator is in contact with environment whose position is xe. xe x••xd f xe If position of manipulator x is x>xe, then the force exerted on the environment is given by (1) Where ke= stiffness of environment System is governed by (2) Where f is input force

  21. STIFFNESS CONTROL OF A SINGLE Degree-of-freedom With xd as shown, it is necessary to show that the PD Control Law. (3) Results in a stable system if the gains are positive. It can also be shown that the steady state force exerted on the environment is (4) has the interpretation of the desired “stiffness of manipulator” Note that for ke large, The control law (3) in trying to eliminate the position error will then cause the force fein (4) to be exerted on the object in the steady state.

  22. Achieving Stiffness Control of an N-dof Manipulator through “Joint Stiffnesses”: Specify a compliance frame {C} & define desired End-Effector stiffness KxR6x6 in this compliance frame. The diagonal elements of Kx have high values for positional control & low values for force control. It is useful to have off-diagonal elements to specify the desired force-motion coupling. • Active research area: see ff. papers: • Ang, “Robots in Industries: Why are They Not So Pervasive?” in Proceedings of Industrial Automation (IA) ’92 Conference, Singapore, 20-23 May ’93. • Schimmds & Peshkin, “Synthesis & Validation of Accom. Matrices for Error Corr. Assy, IEEE Int’l Conf of Rob & Autom. 1990, pp. 714-719.

  23. Achieving Stiffness Control of an N-dof Manipulator through “Joint Stiffnesses”: Since X=J(q)  q and =JT(q)F We have Joint Stiffness Matrix Joint torques The control law to be implemented is =JTKxJ (qd -q) + b

  24. Achieving Stiffness Control of an N-dof Manipulator through “Joint Stiffnesses”: The additional term bcan be used to provide additional damping for stability, or to compensate for say, gravitational & frictional torques. Need not be compensated if motion is slow. In steady state, Force on environment is equal to as desired.

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