### The Textbook

B. Siciliano, L. Sciavicco, L. Villani, G. Oriolo, *Robotics: Modelling, Planning and Control*, Springer, London, 2009, ISBN 978-1-84628-642-1

- Chapter 4 ─ Trajectory Planning
- Chapter 5 ─ Actuators and Sensors
- Chapter 8 ─ Motion Control
- Chapter 9 ─ Force Control
- Chapter 10 ─ Visual Servoing
- Chapter 11 ─ Mobile Robots

### Trajectory Planning

**Unit 1: Joint Space Trajectories**

**Unit 2: Operational Space Trajectories**

*Textbook: Chapter 4 + Section 7.7*

### Unit 1: Joint Space Trajectories

### Unit 1 in a Nutshell

- The difference between path and trajectory is explained
- Techniques for generation of point-to-point motion are presented
- Techniques for generation of motion through a sequence of points are presented
- A technique for automatic scaling of trajectories accounting for dynamic constraints is illustrated

### Path and Trajectory

Generation of suitably smooth trajectories

**Path**: locus of points in the joint space, or in the operational space, which the manipulator has to follow in the execution of the assigned motion (geometric description)**Trajectory**: a path on which a timing law is specified (e.g. velocities and/or accelerations)

**Trajectory planning algorithm**

- Inputs

Path description

Path constraints

Constraints imposed by manipulator dynamics

- Output

Joint (end-effector) trajectories in terms of a time sequence of the values attained by position, velocity and acceleration

### Path and Trajectory II

Reduced number of parameters

- Path
- Extremal points
- Possible intermediate points
- Geometric primitives interpolating the points
- Timing law
- Total trajectory time
- Velocity and/or acceleration at given points

Trajectory planning in the operational space

- Natural task description
- Path constraints
- Singularities
- Redundancy

Trajectory planning in the joint space

- Inverse kinematics
- Control action

### Joint Space Trajectories

Generation of a function interpolating the given vectors of joint variables at each point, in respect of the imposed constraints

- Generated trajectories not very demanding from a computational viewpoint
- Joint positions and velocities (and accelerations) as continuous functions of time
- Undesirable effects minimized (nonsmooth trajectories)

**Point-to-point motion**

- Extremal points and total time

**Motion through a sequence of points**

- Extremal points, intermediate points and transition times

### Point-to-Point Motion

Generation of to move from to in a time

- Cubic polynomial

Computation of coefficients

- Quintic polynomial

### Trapezoidal Velocity Profile II

- specified ()

Trajectory

### Trapezoidal Velocity Profile III

- specified

### Motion Through a Sequence of Points

Opportunity to specify intermediate points (*sequence of points*)

Given path points, find an interpolating function across these points

- -order polynomial

It is not possible to assign the initial and final velocities

As the order of a polynomial increases its oscillatory behaviour increases (nonsmooth trajectories)

Numerical accuracy for computation of polynomial coefficients decreases as order increases

The resulting system of constraint equations is heavy to solve

Polynomial coefficients depend on all the assigned points if it is desired to change a point, all of them have to be recomputed

### Motion Through a Sequence of Points II

Sequence of low-order interpolating polynomials continuous at path points

- Arbitrary values of are imposed at path points
- The values of at path points are assigned according to a certain criterion
- The acceleration has to be continuous at path points

Sequence of interpolating polynomials of order less than three which determine trajectories passing nearby path points at given instants of time

### Interpolating Polynomials with Continuous Accelerations at Path Points (Splines)

- equations in unknowns (fourth-order polynomials for first and last segment?)
*virtual points*(continuity on position, velocity and acceleration) cubic polynomials

### Splines II

- equations for intermediate points

- equations for the initial and final points

- equations for the virtual points

- System of equations in unknowns (coefficients of the cubic polynomials

### Splines III

- Computationally efficient algorithm

- unknowns:

### Splines IV

- variables for given
- Continuity on and
- Continuity on for
- and given
- Continuity on for
- and given

- System of linear equations

### Dynamic Scaling of Trajectories

A technique for trajectory dynamic scaling is introduced, which adapts trajectory planning to the dynamic characteristics of the manipulator

### Dynamic Scaling of Trajectories II

- Time scaling

- Simple choice

Joint corresponding to the largest violation:

### Unit 2: Operational Space Trajectories

### Unit 2 in a Nutshell

- The path primitive concept is introduced to plan position trajectories
- The angle/axis representation is adopted to plan orientation trajectories

### Position Trajectories

- Rectilinear path

### Sequence of Points

- Sequence of points connected by segments

- Via points

### Orientation Trajectories

- Interpolation on the unit vectors (?)

- Interpolation on Euler angles

### Orientation Trajectories II

- Adoption of angle and axis representation ()

### Risorse della lezione

- Trajectory Planning
- Quiz: Robotics Foundations II - Robot Control - lez. #1
- Motion Control
- Quiz: Robotics Foundations II - Robot Control - lez. #2
- Joint Space Decentralized Control
- Quiz: Robotics Foundations II - Robot Control - lez. #3
- Indirect Force Control
- Quiz: Robotics Foundations II - Robot Control - lez. #4
- Vision for Control
- Quiz: Robotics Foundations II - Robot Control - lez. #5
- Parameter Estimation
- Quiz: Robotics Foundations II - Robot Control - lez. #6
- Operational Space Control
- Quiz: Robotics Foundations II - Robot Control - lez. #7
- Visual Servoing
- Quiz: Robotics Foundations II - Robot Control - lez. #8
- Mobile Robots Modelling
- Quiz: Robotics Foundations II - Robot Control - lez. #9
- Direct Force Control
- Quiz: Robotics Foundations II - Robot Control - lez. #10
- Mobile Robots Planning and Control
- Quiz: Robotics Foundations II - Robot Control - lez. #11
- Joint Space Centralized Control
- Quiz: Robotics Foundations II - Robot Control - lez. #12

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