Notes of Manipulation class

发表于 | 更新于: | 分类于 | 字数统计: 844 | 阅读次数:

Matlab Coding

输出

  • fprintf('...')：输出字符；

提取数组、矩阵中元素

1
$ a = A(1); b = B(:,1)

矩阵运算：点乘、逆、转置、行列式

1
2
3
4
$ ans = dot(a,b)
$ ans = inv(T)
$ ans = T'
$ ans = det(T)

绘图

  • plot(x[,y][,'r'])：直接绘制图像；
  • subplot(221): 子窗口绘制图像；
  • title('helloworld'): 设置标题；
  • legend('1','2','3'): 按绘制顺序给图像标注；
  • axis equal; axis([0 5 0 5]): 设置轴刻度范围
  • figure('NumberTitle', 'off', 'Name', 'abc')：隐藏标题号码，设置标题；

其他

  • tic, ...(执行程序), toc：计算程序执行时间；
  • edit file.m：打开文件；

2DPose

Positon and Pose

  • How to describe a point and pose?

Relative Positions

  • How to transform vector from one frame to another?

Relative Poses

  • How to calculate the relative points or poses?

  • Pose algebra

Describing rotation

  • How to calculate the rotation?

  • Rotation matix

Describing rotation and translation

  • Homogeneous transform

Coding Part

  • rot2()：2 dimensional rotation matrix

    1
    2
    rot2(0.2) 
    rot2(30, 'deg')

  • trot2()：homogeneous 2 dimensional rotation matrix

    1
    trot2(30, 'deg')

  • transl2()： homogeneous transformation representing pure translation

    1
    transl2(x,y)

  • se2()： homogeneous transform matrix: providing the translation in the x and y directions as well as the angle to be rotated

    1
    se2(x,y,angle,'deg')/se2(x,y,radian)

    actually: se2(x,y,angle,'deg') == transl2(x,y) * trot2(angle,'deg')

  • e2h() && h2e()： function e2h converts Euclid-ean coordinates to homogeneous and h2e performs the inverse conversion;

    1
    $ p1 = h2e(inv(T1) * e2h(P)) % P = [1;2]

    More compactly this can be written as

    p1 = homtrans( inv(T1), P)

  • Example1: pose compounding is not commutative(交换的)

    1
    2
    3
    4
    T1 = se2(1,2,30,'deg')
    T2 = transl2(2,1) * trot2(0)
    T3 = T1 * T2    %This can be thought of the pose 2 with respect to the frame 1
    T4 = T2 * T1

  • Example2：point with respace to 1

    1
    2
    3
    p = [3 2]'
    plot_point(p)
    p1 = inv(T1) * [p;1] % = p1 = inv(T1) * e2h(p)

3DPose

Basic concept

  • Point && Pose

  • The Right-Hand Rule

    Angles increase positively in the anti-clockwise direction

    Relative Poses

  • Relative position/pose

  • Pose algebra

Representing Orientation in 3-Dimensions

Orthonormal Rotation Matrix

  • 3×3 orthonormal matrix

  • Coding part

    rotx()：3 dimensional rotation matrix

    1
    2
    $ R = rotx(pi/2)
    $ R = rotx(45, 'deg')

    trplot() && tranimate()：display a 3d transformation

    1
    2
    $ trplot(R)
    $ tranimate(R)

Three-Angle Representations

  • Rotation sequences

  • Euler angles：Euler’s rotation theorem requires successive rotation about three axes such that no two successive rotations are about the same axis.

    Representation

    Coding:

    if θ is negative

    if θ=0

  • Cardan angles/Roll-Pitch-Yaw angles

    Representation

    Coding

  • Fundamental problem：Singularities and Gimbal Lock

    This occurs when the rotational axis of the middle term in the sequence becomes parallel to the rotation axis of the first or third term.

Two Vector Representation

  • Two Vector Representation

Rotation about an Arbitrary Vector

  • Any two independent orthonormal coordinate frames can be related by a single rotation about some axis.

  • Finding the axis;

  • Coding part: tr2angvec(R)、eig(R)、angvec2r(pi/2, [1 0 0])



Quaternion

  • The quaternion is an extension of the complex number – a hyper-complex number – and is written as a scalar plus a vector


  • Coding part：Quaternion() is a class

Combining Translation and Orientation

Representing pose

  • Pose

Vecotr-Quaternion

  • Vecotr-Quaternion

homogeneous transformation matrix

  • Form

  • Properties

  • Coding part：transl(x,y,z)、trots(pi/20)

Summary

  • Warpping up

Coding part

  • rotx()：3 dimensional rotation matrix

    1
    2
    $ R = rotx(pi/2)
    $ R = rotx(45, 'deg')

  • trplot() && tranimate()：display a 3d transformation

    1
    2
    $ trplot(R)
    $ tranimate(R)

Time and Motion

Trajectories

  • An important characteristic of a trajectory is that is smooth – position and orientation vary smoothly with time

Smooth One-Dimensional Trajectories

  • Polynomial function of time


  • tpoly(): generates a quintic polynomial trajectory

  • lspb：linear segment (constant velocity) with parabolic blends

Multi-Dimensional Case

  • mtraj()/jtraj(): extend the smooth scalar trajectory to the vector case(多维)

Multi-Segment Trajectories

  • In robotics applications there is often a need to move smoothly along a path through one or more intermediate or via points without stopping. This might be to avoid obstacles in the workplace, or to perform a task that involves following a piecewise continuous trajectory.

  • mstraj()： generates a multi-segment multi-axis trajectory based on a matrix of via points

Interpolation of Orientation in 3D

  • A rotation matrix must be an orthogonal matrix

  • Quaternion interpolation

  • jtraj()/mtraj()： roll-pitch-yaw angles can be interpolated

  • interp()

Cartesian Motion

Another common requirement is a smooth path between two poses in SE(3) which involves change in position as well as in orientation. In robotics this is often referred to as Cartesian motion.

  • Coding part 2019/4/13 10:36:21 2019/4/13 10:36:24

------ 本文结束 ------