第二讲
Link & Joint
Link: rigid-body connected in sequence
Joint: connectors between links
DoF: The degree of freedom. The number of independent parameters that define its configuration.
Notation
add a superscript symbols to denote the recording frame(坐标系), e.g.
Rigid Transformations
the pose of the rigid object:
How to transform so that it overlaps with ?
- First rotate by to align the axes.
- And then translate by to align and .
for coordinate transformation.
Assume a point on the body. Since moves along the body, its coordinate recorded in , denoted as , should never change.
transforms any point in the whole space by the following equation
Homogenous Coordinates
Coordinate transformation under linear form:
Some Rules of Homogenous Coordinate Transformation
Composition rule
Change of observer’s frame:
Joint Categorization
- revolute joints (radians)
- prismatic joints (meters)
- Helical (螺旋的) joint
- Spherical and Socket joint (3DoF)
Base Link
The 0-th link of the robot
Regarded as the “fixed” reference
The spatial frame is attached to it.
End-Effector Link
The last link
e.g. the gripper
A frame is attached to it.
2D Rotation Matrix
将一个点绕原点逆时针旋转一个角度 .
Kinematics Configuration
parameterize the pose of each joint:
- using the relative angle and translation between adjacent frames.
2 representations of the pose of the end-effector:
- Joint space: 每个关节的旋转（轴角表示法）
- Cartesian space: the rigid transformations of the end-effector by .
Forward Kinematics
- Map each coordinate of the joint space to a transformation matrix .
- Calculated by composing transformations along the kinematic chain.
Inverse Kinematics
When one of the following conditions is met, a 6-DoF kinematic structure has a closed-form inverse solution.
- The axes of three consecutive rotational joints intersect at a single point.
- The axes of three consecutive rotational joints are parallel.
这里的 axes 指的是轴角表示法的轴。
Note that this is only a sufficient condition but not necessary.
封闭解：可以用有限的代数运算和常见初等函数明确写出解析解
and
: Special Orthogonal Group.
3DoF.
: Special Euclidean Group
6DoF.
Euler Angle
Yaw, Pitch and Roll.
Euler angle is not unique for some rotations.
Angle-Axis Representation
Any rotation is equivalent to a rotation about a fixed axis through a positive angle .
Skew-Symmetric Matrix
Skew-symmetric matrix operator
so that
For any
i.e.
Taylor’s expansion