机器人学基础_第3章_机器人运动学

……
3.1 Representation of Kinematics Equation of Manipulator
8
3.1 Representation of Kinematics Equation of Robot Manipulator
T矩阵：A矩阵的乘积 。 对于六连杆机械手，有下列T矩阵 ：
Kinematics treats motion without regard to the forces that cause it. Within the science of kinematics one studies the position, velocity, acceleration, and all higher order derivatives of the position variables (with respect to time or any other variable). 从几何学 几何学的观点来处 几何学 理手指位置 手指位置P与关节变量 关节变量 手指位置 L1, L2, θ1 和 θ 2的关系称为 运动学(Kinematics)。 运动学
3.0 Introduction to Robot Kinematics
2
3.0 Introduction to Robot Kinematics In manipulator robotics, there are two kinematics tasks: Direct (also forward) kinematics – Given are joint relations (rotations, translations) for the robot arm. Task: What is the orientation and position of the end effector? Inverse kinematics – Given is desired end effector position and orientation. Task: What are the joint rotations and orientations to achieve this?
Cyl ( z , α , r ) = Trans (0,0, z ) Rot ( z , α )Trans ( r ,0,0)
图3.4 用柱面坐标和球面坐标表示位置
3.1 Representation of Kinematics Equation of ManipKinetic Position and Coordinate
3.1 Representation of Kinematics Equation of Manipulator
12
3.1.1 Kinetic Pose and Oriented Angle Roll, Pitch, Yaw to represent motion pose
另一种常用的旋转集合是横滚（roll）、俯仰（pitch） 和偏转（yaw）。
Description in Spherical Coordinates 用球面坐标表示手臂运动位置矢量的方法，对应于 沿 z 轴平移 r，再绕 y 轴旋转β角，最后绕 z 轴旋 转 α 角，如图3.4(b)所示，即为：
Sph(α , β , r ) = Rot ( z , α ) Rot ( y , β )Trans (0,0, r )
3.1 Representation of Kinematics Equation of Manipulator
7
3.1 Representation of Kinematics Equation of Robot Manipulator
机械手是一系列由关节连接起来的连杆构成的。为 机械手的每一连杆建立一个坐标系，并用齐次变换来 描述这些坐标系间的相对位置和姿态。 A矩阵：一个描述两连杆间坐标系相对关系的齐次变 换 ，如；各 A 矩阵的乘积称为 T 矩阵 。 例如： A1，A2，A3 T1=A1 T2=A1A2 T3=A1A2A3
(3.9)
3.1 Representation of Kinematics Equation of Manipulator
17
3.1 Representation of Kinetic Equation of Robot Manipulator
4
Example of Inverse Kinematics
θ2 = π − α
y L2 sinθ2 θ1 = arctan( ) − arctan( ) x L1 + L2 cosθ2
式中
2 −(x2 + y2 ) + L1 + L2 2 α = arccos 2L1L2
T6 = A1 A2 A3 A4 A5 A6
(3.1)
一个六连杆机械手可具有六个自由度，每个连 杆含有一个自由度，并能在其运动范围内任意 定位与定向。
3.1 Representation of Kinematics Equation of Manipulator
9
3.1 Representation of Kinematics Equation of Robot Manipulator
图3.3 用横滚、俯仰和偏转表示机械手运动姿态
3.1 Representation of Kinematics Equation of Manipulator
13
3.1.1 Kinetic Pose and Oriented Angle
对于旋转次序，规定：
RPY (ϕ ,θ ,ψ ) = Rot ( z ,ϕ ) Rot ( y,θ ) Rot ( x,ψ )
图3.1 矢量n,o,a和p
10
3.1 Representation of Kinematics Equation of Manipulator
3.1.1 Kinetic Pose and Oriented Angle
因此，变换T6具有下列元素（同式2.35）。
nx n T6 = y nz 0 ox oy oz 0 ax ay az 0 px py pz 1
机器人学基础
第三章 机器人运动学
Fundamentals of Robotics
Ch.3 Kinematics of Robots
中南大学 蔡自兴， 蔡自兴，谢 斌 zxcai, xiebin@ 2010
Fundamentals of Robotics
1
3.0 Introduction to Robot Kinematics
同样，如果用向量表示上述关系式，其一般可表示为
θ = f (r )
−1
3.0 Introduction to Robot Kinematics
5
Example of Inverse Kinematics 机器人到达给定的手爪位置 P 有两个姿态满足要求，即图中 的 α ′ = −α 也是其解。此时 θ1 和 θ 2 变成为另外的值，即逆运 逆运 动学的解不是惟一的。 动学的解不是惟一的 将运动学公式 r = f (θ ) 两边微分即可得到机器 人手爪的速度和关节速度的关系，再进一步进行微 分将得到加速度之间的关系，处理这些关系也是机 器人的运动学问题。
3.0 Introduction to Robot Kinematics
6
3.1 Representation of Kinematics Equation of Robot Manipulator
Mechanics of a manipulator can be represented as a kinematics chain of rigid bodies (links) connected by revolute or prismatic joints. One end of the chain is constrained to a base, while an end effector is mounted to the other end of the chain. The resulting motion is obtained by composition of the elementary motions of each link with respect to the previous one.
3.1.1 Kinetic Pose and Oriented Angle 运动姿态和方向角 Motion Direction
原点由矢量p表示。 approach vector a：z向矢量 orientation vector o：y向矢量 normal vector n：x向矢量， Forming a right-hand frame： n = o × a or a = n × o
3.0 Introduction to Robot Kinematics
3
Example of Direct Kinematics
Define position of end effector and the joint variable，
x r= y
θ1 θ = θ2
(3.2)
六连杆机械手的T 矩阵（ T6 ）可由指定其16个元素的数值 来决定。在这16个元素中，只有12个元素具有实际含义。
3.1 Representation of Kinematics Equation of Manipulator
11
3.1.1 Kinetic Pose and Oriented Angle Euler angle to represent motion pose
(3.6)
3.1 Representation of Kinematics Equation of Manipulator