Robotics & Automation Final

A coordinate frame is defined by

Three orthogonal unit vectors ;

A vector’s value changes when

The coordinate frame / magnitude / direction changes (all of the above) ;

A transformation between two frames expresses

The relationship between the frames ;

An orthonormal basis satisfies

i·j = 0 and |i| = 1 ;

A rotation matrix is

A 3×3 matrix with orthogonal columns of unit length ;

For a valid rotation matrix R

RᵀR = I and det(R) = 1 ;

The inverse of a rotation matrix equals

Its transpose ;

The standard rotation matrix about the z-axis by angle θ is

[[cosθ −sinθ 0]; [sinθ cosθ 0]; [0 0 1]] ;

The result of successive rotations is obtained by

Multiplying rotation matrices in order ;

Rotating a coordinate frame vs rotating a vector leads to

The transpose operation ;

A position vector represents

The location of a point relative to a reference frame ;

Rigid body motion preserves

Distances and angles ;

Rigid body motion is described by

A rotation and a translation ;

The composition of two rigid transformations is

Another rigid transformation ;

The homogeneous transformation matrix has size

4×4 ;

A 3D point in homogeneous coordinates is represented as

[x y z 1]ᵀ ;

The point transformation in homogeneous form is

p′ = T p ;

If T₀¹ and T₁² are known then T₀² is

T₀¹ T₁² ;

The homogeneous transformation T represents

Both orientation and position ;

The Denavit–Hartenberg (DH) method is used to

Represent geometric relationships between links and joints ;

Each link in the DH convention is characterized by

a

In the DH convention

the z-axis of each frame is aligned with

For a revolute joint

the DH parameter that varies is

A typical cylindrical robot joint sequence is

RPP ;

The workspace of a cylindrical robot is

A cylindrical volume ;

A spherical wrist allows

Orientation control independent of position ;

In a spherical wrist

the three joint axes

Forward kinematics determines

End-effector pose from joint variables ;

Forward kinematics is computed by

Multiplying transformation matrices ;

Inverse kinematics involves

Finding joint variables for a desired end-effector pose ;

Inverse kinematics generally has

Multiple possible solutions ;

A skew-symmetric matrix satisfies

Sᵀ = −S ;

The diagonal elements of a skew-symmetric matrix are

0 ;

The skew-symmetric matrix for ω = [ωₓ ωᵧ ω_z]ᵀ is

[[0 −ωz ωᵧ]; [ωz 0 −ωₓ]; [−ωᵧ ωₓ 0]] ;

S(ω)v = ω × v expresses

The cross product using a matrix operator ;

Angular velocity describes

Rate of change of orientation over time ;

Angular velocity is represented as

A vector ;

The matrix S(ω) represents

Instantaneous angular velocity ;

For a point P

linear velocity is

A rigid body’s instantaneous motion is

Both linear and angular velocities ;

The Jacobian matrix relates

Joint velocities to end-effector velocities ;

For a 6-DOF manipulator

the Jacobian size is

The Jacobian is divided into

Translational and rotational parts ;

Each column of the Jacobian represents

The effect of one joint’s motion on end-effector velocity ;

The Jacobian represents

Instantaneous velocity mapping between joint and task space ;