Kinematics

The Robot System Kinematics

Hardware Mechanical Design Actuators

Dynamics Control System Task Planning

Software

Sensors

Robot Kinematics. – In order to control and programme a robot we must have knowledge of both it’s spatial arrangement and a means of reference to the environment. – KINEMATICS - the analytical study of the geometry of motion of a robot arm: with respect to a fixed reference coordinate system ■ without regard to the forces or moments ■

z

Co-ordinate Framesx

y Right-handed Co-ordinate frame

Camera Frame x

Tool Frame x

Link Frame x

Goal Frame x

Base x Frame

Kinematic Relationship – Between two frames we have a kinematic relationship - basically a translation and a rotation. z

y x

x y

z

– This relationship is mathematically represented by a 4 × 4 Homogeneous Transformation Matrix.

Homogeneous Transformations 3×3 Rotational Matrix

r1 r2 r3

∆x

r4 r5 r6

∆y

r7 r8 r9

∆z

0

1

0

0

1 × 3 Perspective

3 × 1 Translation

Global Scale

Kinematic Considerations Using kinematics to describe the spatial configuration of a robot gives us two approaches: ■ Forward Kinematics. (direct) ■

– Given the joint angles for the robot, what is the orientation and position of the end effector? ■

Inverse Kinematics. – Given a desired end effector position what are the joint angles to achieve

Inverse Kinematics For a robot system the inverse kinematic problem is one of the most difficult to solve. ■ The robot controller must solve a set of non-linear simultaneous equations. ■ The problems can be summarised as: ■

– The existence of multiple solutions. –

Multiple Solutions Goal

• This two link planar manipulator has two possible solutions. • This problem gets worse with more ‘Degrees of Freedom’. • Redundancy of movement.

Non Existence of Solution Goal • A goal outside the workspace of the robot has no solution.

• An unreachable point can also be within the workspace of the manipulator - physical constraints. • A singularity is a place of ∞ acceleration - trajectory tracking.

Kinematics → Control ■

Kinematics is the first step towards robotic control.

Cartesian Space z

y x

Joint Space

Actuator Space

Joint Space Trajectories For a robot to operate efficiently it must be able to move from point to point in space. ■ A trajectory is a time history of position, velocity and acceleration for each joint. ■ Trajectories are computed at run time and updated at a certain rate - the Path Update Rate. (PUMA robot ■

Joint Space Trajectory Planning (θ0 , t0) A

Consider a robot with only one link. •Kinematics gives one configuration for B. B (θf , tf)

•Choice of two trajectories to get there. •May wish to specify a via point - maybe to avoid an obstacle.

Joint Space Schemes. We need to describe path shapes in terms of functions of joint angles. θ(t) angle ■

θf

Lots of choices for continuous functions

θ0 0

tf

time

Cubic Polynomials – To move a single revolute joint from A to B in a given time gives four constraints. • A starts at rest and at angle θ0 θ ( 0 ) = θ0 θ ( 0 ) = 0 •

θ (t f ) = θ f θ (t f ) = 0

B finishes at rest and at angle θf

A cubic polynomial has four co-efficients which satisfy the four constraints:

θ ( t ) = a0 + a1t + a2t + a3t 2

3

An Exercise for you: ■

Place the initial constraints into the formulae for position, velocity and acceleration and prove that the coeffecients are:

a0 = θ0

a1 = 0

3 a2 = 2 (θ f − θ0 ) tf

2 a3 = − 3 (θ f − θ0 ) tf

An exercise for us – Given a single link robot arm with a revolute joint. Construct a cubic path function to take it from it’s present rest at 10 degrees to finish at rest at a desired end position of 110 degrees.

a0 = θ0

a1 = 0

3 a2 = 2 (θ f − θ0 ) tf

2 a3 = − 3 (θ f − θ0 ) tf

Making A Spline. ■

A via point gives a constraint with •

•

θ ( tvia ) = θvia

angle Via points

B

A

tvia1

tvia2

time

More Joint Space Schemes ■

Quintic Polynomials. – The cubic polynomial does not specify accelerations at the start and end of the motion. This adds two more constraints which can only be represented by a quintic polynomial. i.e. a5t5

■

Linear Functions with parabolic Blends. – Linear function requires an infinite acceleration to get it started so parabolic

Kinematics → Control ■

Kinematics is the first step towards robotic control.

Cartesian Space z

y x

Joint Space

Actuator Space