Forward Kinematics

ME/ECE 439 2007 Professor N. J. Ferrier

Forward Kinematics

Professor Nicola FerrierME Room 2246, 265-8793

[email protected]


Forward Kinematics• Modeling assumptions• Review:

– Spatial Coordinates• Pose = Position + Orientation

– Rotation Matrices

– Homogeneous Coordinates

• Frame Assignment– Denavit Hartenberg Parameters

• Robot Kinematics– End-effector Position,– Velocity, & – Acceleration

Today

Next

Lecture


Industrial Robot

sequence of rigid bodies (links) connected by

means of articulations

(joints)


Robot Basics: Modeling

• Kinematics:– Relationship between

the joint angles, velocities & accelerations and the end-effector position, velocity, & acceleration


Modeling Robot Manipulators• Open kinematic chain (in this course)

• One sequence of links connecting the two ends of the chain (Closed kinematic chains form a loop)

• Prismatic or revolute joints, each with a single degree of mobility

• Prismatic: translational motion between links• Revolute: rotational motion between links

• Degrees of mobility (joints) vs. degrees of freedom (task)

• Positioning and orienting requires 6 DOF• Redundant: degrees of mobility > degrees of freedom

• Workspace • Portion of environment where the end-effector can

access


Modeling Robot Manipulators

• Open kinematic chain – sequence of links with one end constrained to

the base, the other to the end-effector

Base

End-effector


Modeling Robot Manipulators

• Motion is a composition of elementary motions

Base

End-effector

Joint 1

Joint 2

Joint 3


Kinematic Modeling of Manipulators

• Composition of elementary motion of each link

• Use linear algebra + systematic approach

• Obtain an expression for the pose of the end-effector as a function of joint variables qi (angles/displacements) and link geometry (link lengths and relative orientations)

Pe = f(q1,q2,,qn ;l1,ln,1,n)


Pose of a Rigid Body

• Pose = Position + Orientation• Physical space, E3, has no natural

coordinates.• In mathematical terms, a coordinate

map is a homeomorphism (1-1, onto differentiable mapping with a differentiable inverse) of a subset of space to an open subset of R3.– A point, P, is assigned a 3-vector:

AP = (x,y,z) where A denotes the frame of reference


A BX

X

Y

Y

Z

Z

AP = (x,y,z)

BP = (x,y,z)

P



• Pose = Position + Orientation

How do we do this?



• Pose = Position + Orientation• Orientation of the rigid body

– Attach a orthonormal FRAME to the body– Express the unit vectors of this frame with

respect to the reference frame

XA

YA

ZA


Rotation Matrices

• OXYZ & OUVW have coincident origins at O– OUVW is fixed to the object

– OXYZ has unit vectors in the directions of the three axes ix, jy,and kz

– OUVW has unit vectors in the directions of the three axes iu, jv,and kw

• Point P can be expressed in either frame:


OXU

V

Y

W

ZAP = (x,y,z)

BP = (u,v,w)

P


OXU

V

Y

W

ZAP = (x,y,z)

P

BP = (u,v,w)


OXU

V

Y

W

ZAP = (x,y,z)BP = (u,v,w)

P


Rotation Matrices


Rotation Matrices

1

X axis expressed wrt Ouvw


Rotation Matrices

1

Y axis expressed wrt Ouvw


Rotation Matrices

1

Z axis expressed wrt Ouvw


Rotation Matrices


Rotation Matrices

Z axis expressed wrt Ouvw

X axis expressed wrt Ouvw

Y axis expressed wrt Ouvw


Rotation Matrices

1

U axis expressed wrt Oxyz


Rotation Matrices

U axis expressed wrt Oxyz

V axis expressed wrt Oxyz

W axis expressed wrt Oxyz


Properties of Rotation Matrices• Column vectors are the unit vectors of the

orthonormal frame– They are mutually orthogonal– They have unit length

• The inverse relationship is:

– Row vectors are also orthogonal unit vectors


Properties of Rotation Matrices

• Rotation matrices are orthogonal

• The transpose is the inverse:

• For right-handed systems

– Determinant = -1(Left handed)

• Eigenvectors of the matrix form the axis of rotation


Elementary Rotations: X axis

X Y

Z


Elementary Rotations: Y axis

X Y

Z


Elementary Rotations: Z-axis

X Y

Z


Composition of Rotation Matrices

• Express P in 3 coincident rotated frames



• Recall for matrices AB BA

(matrix multiplication is not commutative)

Rot[Z,90] Rot[Y,-90]



• Recall for matrices AB BA

(matrix multiplication is not commutative)

Rot[Z,90]

Rot[Y,-90]


Rot[Z,90]Rot[Y,-90]

Rot[Z,90] Rot[Y,-90]


Rot[z,90]Rot[y,-90] Rot[y,-90] Rot[z,90]


Decomposition of Rotation Matrices

• Rotation Matrices contain 9 elements• Rotation matrices are orthogonal

– (6 non-linear constraints)

3 parameters describe rotation

• Decomposition is not unique



• Euler Angles

• Roll, Pitch, and Yaw



• Angle Axis



• Angle Axis

• Elementary Rotations



• Pose = Position + Orientation

Ok. Now we know what to do about

orientation…let’s get back to pose


Spatial Description of Body

• position of the origin with an orientation

AX Y

Z

B


Homogeneous Coordinates

• Notational convenience


Composition of Homogeneous Transformations

• Before:

• After



• Inverse Transformation



• Inverse Transformation

Orthogonal: no matrix inversion!

Forward Kinematics

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