1. Controls

2. Given a set of desired Tool frame positions and orientations (that trace out a path over time, t ), there will be a corresponding required set of desired joint angles and joint offsets:  i (t): desired joint angles and, D i (t): desired joint offsets To obtain perfect path tracking, it is required that  i (t) =  i (t) and d i (t) = D i (t) And also, must match velocities and accelerations: angular velocity:  i (t) =  i (t) angular acceleration:  i (t) =  i (t) translational velocity:d i (t) = D i (t)translational acceleration: d i (t) = D i (t) actual (measured) joint angles = desired joint angles actual (measured) joint offsets = desired joint offsets prismatic revolute......

3. Forces and torques are applied (by actuators), respectively, to position prismatic joints and to orient revolute joints. m F J  F: force  : torque m: mass (translational inertia) J: moment of inertia (rotational inertia) Newton’s second law of motion relates forces/torques to accelerations: translational:F = m · d rotational:  = J · ..

4. Ideally, for perfect tracking, apply the following actuation: F = m · D  = J ·  to each joint. However, in practice, the presence of disturbances and nonlinearities (in robot and in workspace) that are hard to characterize lead to the use of closed loop feedback control. To introduce the idea, consider the task of steering a car along a curved road. Open-loop control.. e = desired course of travel – actual course of travel Output of car (actual course of travel) Desired course of travel road car e

5. To stay on path of road: two directions sensed and then car’s steering wheel is adjusted appropriately Actual course of travel (output) Desired course of travel (reference input) driver controller Steering mechanism actuator car plant eyes sensor + - e

6. For a typical robot joint: The microcontroller may be programmed to execute various control algorithms. Simplest possibility: proportional (P) control.  (t) or D(t) Micro- controller  (t) or d(t) Motor F or t Robot joint plant encoder sensor + - e 1 2 1 2 ADC DAC

7. For a typical robot joint: The microcontroller may be programmed to execute various control algorithms. Simplest possibility: proportional (P) control.  (t) or D(t) Micro- controller  (t) or d(t) Motor F or t Robot joint plant encoder sensor + - e 1 2 1 2 ADC DAC