1. Trajectory Generation
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

2. Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

3. Trajectory Generation – Problem Definition
Given: Manipulator geometry, End Effector
via point or Trajectory
Compute: The trajectory of each joint such that
the end effector move in space
from point A to Point B
Solution (Domains)
Joint space / Cartesian Space
Definitions
Trajectory (Definition) - Time history of position, velocity, and acceleration for each DOF.
Trajectory Generation – Methods of computing a trajectory that describes the desired motion of a manipulator in a multidimensional space
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

4. Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

5. Trajectory Generation – Problem Definition
Human Machine interface (requirements)
Human - Specifying trajectories with simple description of the desired motion
Example – start / end points position and orientation of the end effectors
System - Designing the details of the trajectory
Example – Design the exact shape of the path, duration, joint velocity, etc.
Trajectory Representation
Representation of trajectory in the computer after they were planned
Trajectory Generation
Generation occurs at runtime (real time) where positions, velocities, and accelerations are computed.
Path update rate Hz
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

6. General Consideration
General approach for the motion of the manipulator
Specify the path as a motion of the tool frame {T} relative to the station frame {S}. Frame {S} may change it position in time (e.g. conveyer belt)
Advantages
Decouple the motion description from any particular robot, end effector, or workspace.
Modularity – Use the same path with:
Different robot
Different tool size
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

7. General Consideration
Basic Problem – Move the tool frame {T} from its initial position / orientation {T_initial} to the final position / orientation {T_final}.
Specific Description
Via Point – Intermediate points between the initial and the final end- effector locations that the end-effector mast go through and match it position and orientation along the trajectory.
Each via point is defined by a frame defining the position/orintataion of the tool with respect to the station frame
Path Points – includes all the via points along with the initial and final points
Point (Frame) – Every point on the trajectory is define by a frame (spatial description)
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

8. General Consideration
“Smooth” Path or Function
Continuous path / function with first and second derivatives.
Add constrains on the spatial and temporal qualities of the path between the via-points
Great verity of ways in which paths might be specified or planned.
Implications of non-smooth path
Increase wear in the mechanism (rough jerky movement)
Vibration – exciting resonances.
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

9. Joint Space Schemes
Joint space Schemes – Path shapes (in space and in time) are described in terms of functions in the joint space.
General process (Steps) given initial and target P/O
Select a path point or via point (desired position and orientation of the tool frame {T} with respect to the base frame {s})
Convert each of the “via point” into a set of joint angles using the invers kinematics
Find a smooth function for each of the n joints that pass trough the via points, and end the goal point.
Note 1: The time required to complete each segment is the same for each joint such that the all the joints will reach the via point at the same time. Thus resulting in the position and orientation of the frame {T} at the via point.
Note 2: The joints move independently with only one time restriction (Note 1)
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

10. Joint Space Schemes – Cubic Polynomials
Define a function for each joint such that value at is the initial position of the joint and whose value at is the desire goal position of the joint
There are many smooth functions that may be used to interpolate the joint value.
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

11. Joint Space Schemes – Cubic Polynomials
Problem - Define a function for each joint such that it value at
is the initial position of the joint and at
is the desired goal position of the joint
Given - Constrains on
What should be the order of the polynomial function to meet these constrains?
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 11

12. Joint Space Schemes – Cubic Polynomials
Solution - The four constraints can be satisfied by a polynomial of at least third degree
The joint velocity and acceleration
Combined with the four desired constraints yields four equations in four unknowns
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

13. Joint Space Schemes – Cubic Polynomials
Solving these equations for the we obtain
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

14. Joint Space Schemes – Cubic Polynomials
Example – A single-link robot with a rotary joint is motionless at degrees. It is desired to move the joint in a smooth manner to degrees in 3 seconds. Find the coefficient of the cubic polynomial that accomplish this motion and brings the manipulator to rest at the goal
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

15. Joint Space Schemes – Cubic Polynomials
The velocity profile of any cubic function is a parabola
The acceleration profile of any cubic function is linear
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

16. Joint Space Schemes – Cubic Polynomials – Via Points
Previous Method - The manipulator comes to rest at each via point
General Requirement - Pass through a point without stopping
Problem - Define a function for each joint such that it value at
is the initial position of the joint and at
is the desire goal position of the joint
Given - Constrains on such that the velocities at the via points are not zero but rather some known velocities
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 16

17. Joint Space Schemes – Cubic Polynomials – Via Points
Solution - The four constraints can be satisfied by a polynomial
Combined with the four desired constraints yields four equations in four unknowns
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 17

18. Joint Space Schemes – Cubic Polynomials – Via Points
Solving these equations for the we obtain
Given - velocities at each via point are
Solution - Apply these equations for each segment of the trajectory.
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 18

19. Joint Space Schemes – Cubic Polynomials – Via Points
Specify the desired velocities at each segment:
User Definition – Desired Cartesian linear and angular velocity of the tool frame at each via point.
System Definition – The system automatically chooses the velocities (Cartesian or angular) automatically using a suitable heuristic method.
System Definition – The system automatically chooses the velocities (Cartesian or angular) to cause the acceleration at the via point to be continuous.
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

20. Joint Space Schemes – Cubic Polynomials – Via Points
User Definition – Desired Cartesian linear and angular velocity of the tool frame at each via point.
Mapping (Cartesian Space to Joint Space) - Cartesian velocities at the via point are “mapped” to desired joint rates by using the inverse Jacobian
Singularity - If the manipulator is at a singular point at a particular via point then the user is not free to choose an arbitrarily velocity at this point.
Difficult
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 20

21. Joint Space Schemes – Cubic Polynomials – Via Points
2. System Definition – The system automatically chooses the velocities (Cartesian or angular) using a suitable heuristic method given a trajectory .
Heuristic method
Consider a path defined by via points
Connect the via points with straight lines
If the slope change sign
Set the velocity at the via point to be zero
If the slope have the same sign
Calculate the average between the to velocities at the via point.
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 21

22. Joint Space Schemes – Cubic Polynomials – Via Points
System Definition – The system automatically chooses the velocities (Cartesian or angular) to cause the acceleration at the via point to be continuous.
Spline - Enforcing the velocity and the acceleration to be continuous at the via point
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 22

23. Cubic Polynomials – Via Points – Example
Solve for the coefficients of two cubic functions that are connected in a two segment spline with a continuous acceleration at the intermediate via point.
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

24. Cubic Polynomials – Via Points – Example
The joint angle velocity and acceleration for each segment (8 unknowns )
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 24

25. Cubic Polynomials – Via Points – Example
Position at the beginning and end of each segment
Segment 1
Segment 2
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 25

26. Cubic Polynomials – Via Points – Example
Velocity at the beginning of the interval
Velocity at the end of the interval
Velocity at the mid point between the intervals
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 26

27. Cubic Polynomials – Via Points – Example
Acceleration at the mid point between the intervals
Solve 8 equations with 8 unknown
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 27

28. Cubic Polynomials – Via Points – Example
Solution for the 8 equations
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 28

29. Joint Space Schemes – High Order Polynomials
Problem - Define a function for each joint such that it value at
is the time at the initial position
is the time at the desired goal position
Given - Constrains on the position velocity and acceleration at the beginning and the end of the path segment
What should be the order of the polynomial function to meet these constrains?
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 29

30. Joint Space Schemes – High Order Polynomials
Solution - The six constraints can be satisfied by a polynomial of at least fifth order
Combined with the six desired constraints yields six equations with six unknowns
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 30

31. Joint Space Schemes – Cubic Polynomials
Solving these equations for the we obtain
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 31

32. Joint Space Schemes – Linear Function With Parabolic Blend
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 32

33. Joint Space Schemes – Linear Function With Parabolic Blend
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 33

34. Joint Space Schemes – Linear Function With Parabolic Blend
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 34

35. Joint Space Schemes – Linear Function With Parabolic Blend
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 35

36. Joint Space Schemes – Linear Function With Parabolic Blend
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 36

37. Joint Space Schemes – Linear Function With Parabolic Blend
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 37

38. Cartesian Space Schemes – Introduction
Joint Space Schemes
Advantages –
Path go through all the via and goal points
Points can be specified by Cartesian frames.
Disadvantages -
End effector moves along a curved line (not a straight line - shortest distance).
Path depends on the particular joint kinematics of the manipulator
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 38

39. Cartesian Space Schemes – Introduction
Advantage
Most common path is straight line (shortest). Other shapes can also be used.
Disadvantage
Computationally expansive to execute – At run time the inverse kinematics needs to be solved at path update rate ( Hz)
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA 39

40. Cartesian Space Scheme – Cartesian Straight Line
General Approach - Define the path (in the Cartesian space) as
straight lines (linear functions)
Parabolic lines (blends)
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

41. Cartesian Space Scheme – Cartesian Straight Line
General Approach (continue)
Every point along the path is defined by position and orientation of the end effector
End Effector Position – Vector – Easy interpolation
End Effector Ordination – Matrix – Impossible to interpolate (interpolating the individual elements of the matrix violate the requirements that all column of the matrix must be orthogonal)
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

42. Equivalent Angle – Axis Representation
Start with the frame coincident with a know frame {A}; then rotate frame {B} about a vector by an angle according to the right hand rule.
Equivalent Angle – Axis Representation
Vector is called the equivalent axis of a finite rotation.
The specification of requires two parameters since it length is always 1.
The angle specify the third parameter
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

43. Equivalent Angle – Axis Representation
Conversion 1 - Conversion for single angle axis representation to rotation matrix representation
Conversion 2 – Compute and given a rotation a matrix
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

44. Cartesian Space Scheme – Cartesian Straight Line
Combining the angle-axis representation of orientation with the 3x1 Cartesian position representation we have a 6x1 representation of Cartesian position and orientation.
Consider a via point specified relative to a station point frame as
Frame {A} specifies a via point
Position of the end effector given by
Orientation of the end effector given by
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

45. Cartesian Space Scheme – Cartesian Straight Line
Convert the rotation matrix into an angle axis representation
Use the symbol to represent 6x1 position and orientation
Where is formed by scaling the unite vector by the amount of rotation

46. Cartesian Space Scheme – Cartesian Straight Line
Process - For a given trajectory we describe a spline function that smoothly vary these six quantities from path point to path point as a function of time.
Linear Spline with parabolic bland
Path shape between via points will be linear
When via points are passed, the linear and angular velocity of the end effector are changed smoothly
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

47. Cartesian Space Scheme – Cartesian Straight Line
Complication – The angle-axis representation is not unique
In going from via point {A} to a via point {B}, the total amount of rotation should be minimized
Choose such that
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

48. Cartesian Space Scheme – Cartesian Straight Line
The splines are composed of linear and parabolic blend section
Constrain
The transition between the linear segment and the parabolic segment for all the DOF must take place at the same time.
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

49. Geometric Problems – Cartesian Paths
Problem Type 1 – Unreachable Intermediate Points
The initial and the final point are in the reachable workspace however some point along the path may be out of the workspace.
Solution
Joint space path – reachable
Cartesian straight Path – Unreachable
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

50. Geometric Problems – Cartesian Paths
Problem Type 2 – High Joint Rate Near Singularity.
In singularity the velocity of one or more joint approach infinity.
The velocity of the mechanism are upper bounded, approaching singularity results in the manipulator’s deviation form the desired path.
Solution
Slow down the velocity such that all the joint velocities will remain in their bounded velocities
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

51. Geometric Problems – Cartesian Paths
Problem Type 3 – Start and Goal reachable in different solutions
Joint limits may restrict the number of solutions that the manipulator may use given a goal point.
Solution
Switch between joint space (default) and Cartesian space trajectories (used only if needed)
Instructor: Jacob Rosen
Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

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Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

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Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

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Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA