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Page 1: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

ROBOT CONTROL

T. Bajd and M. Mihelj

Page 2: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Robot control deals with computation of the forces or torques which must be generated by the actuators in order to successfully accomplish the robot task.

• The robot task can be – execution of the motion in a free space, where position control

is performed, or – in contact with the environment, where control of the contact

force is required.• The choice of the control method depends on

– the robot task,– the mechanical structure of the robot mechanism.

• Robot control usually takes place in the world coordinate frame, which is defined by the user and is called also the coordinate frame of the robot task.

Robot control

Page 3: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

General control approach

End-effector pose

Position Orientation

RPY notation of the orientation

Page 4: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Control loop is closed separately for each particular degree of freedom

• Less suitable for robotic systems characterized by nonlinear and time varying behavior

• Position error computation– Reference positions– Measured robot joint positions– Position error

PD position control

Page 5: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Control law; computation of control variable (torque, velocity)

• Actuation of robot motors is proportional to the error• Velocity feedback loop introduces damping into the system• Velocity error can be introduced into the control law (faster

system response)

• leading to

PD position control

Page 6: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

Block schemes

Page 7: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Robot inverse dynamic model

• In static conditions can be simplified to

• Estimated gravity term part of the control law

PD position control with gravity compensation

Page 8: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

PD position control with gravity compensation

Page 9: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Robot inverse dynamic model

• Robot forward dynamic model

• Define new variable

• leading to

Robot dynamic model

Page 10: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Assume that the robot dynamic model is known– inertial matrix is an approximation of real values ,– represents an approximation of

• Consider the following control law

• where input y will be defined later.

Inverse dynamics control

Page 11: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

Inverse dynamics control block scheme

y represents computed acceleration in joint space

Page 12: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• position error

• velocity error

• control law

• error dynamics

PD position control

Page 13: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

Controller block scheme

Page 14: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Definition of pose error

• Control based on the transposed Jacobian matrix

• Control based on the inverse Jacobian matrix

Robot control in external coordinates

Page 15: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• End-effector position

• Differential kinematics

Manipulator Jacobian matrix

Jacobian matrix

Page 16: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Jacobian matrix

Manipulator Jacobian matrix

Page 17: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Inverse velocity relation

• For a square matrix of dimension two

• Inverse velocity relation

• Inverse Jacobian matrix equals

Inverse Jacobian matrix

Page 18: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Robot in contact with the environment (contact force f)• Find resulting joint torques

• In matrix form

Transposed Jacobian matrix

Page 19: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Velocity relation

• Force/torque relation

• Transposed matrix

• Force/torque relation

Transposed Jacobian matrix

Transposed Jacobian matrix

Page 20: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Pose error in external coordinates

• Control law formulation (control variable in external coordinates)

• Control variable in joint space

Transposed Jacobian matrix based Control

Page 21: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Velocity relation

• Relation for small displacements

• Relation for small pose errors

• Control law in joint space

Inverse Jacobian matrix based Control

Page 22: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

PD position control with gravity compensation in external coordinates• Control law

Page 23: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Inverse dynamics

• Velocity relation

• Acceleration relation

• Computed acceleration for external coordinates control

Inverse dynamics control in external coordinates

Page 24: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Pose error

• Velocity error

• Acceleration error

• Error dynamics

• Control law

Inverse dynamics control in external coordinates

Page 25: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

Inverse dynamics control in external coordinates – block scheme

Page 26: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• control of end-effector desired pose while the robot is in contact with the environment

– case of robot assembly (inserting a peg into a hole)

– robot movement assures minimal contact force during action

• robot end-effector exerts a predetermined force on the environment

– case of machining parts with robot (grinding)

– robot movement depends on the difference between the desired and the actual contact force.

Control of contact force with environment

Page 27: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Dynamic model with contact force

• Define new variable

• leading to

Robot dynamics with contact

result of interaction with the environment

Page 28: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Inverse dynamics with contact

• Forward dynamics with contact

• Control law

Inverse dynamics control with contact

Page 29: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Force control is based on position control

• Reference values for acceleration, velocity and pose are computed from force error

Force control

Page 30: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Force error

• Predefined manipulator behavior via inertia and damping matrices and

• Reference trajectory

Force control

Page 31: ROBOT CONTROL

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010

• Parallel composition assumes force control in certain direction and pose control in other directions

Parallel composition


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