R1. Rotations and homogenous transformations

Download the Robotic toolbox package from the link on laboratory page. Unzip the package in your matlab tollbox directory (e.g ~/matlab/toolbox).

• Choose a value of an angle θ
• Write a formula for a rotation matrix R in planar case (\mathcal{R}²)
• Define matrix R1 in Matlab

  >>  R1=rot2( theta )
  

  check transpose, determinant and inverse of R1.

• Define matrix R2 for a different angle and verify properties of R2.
• Calculate the product of R1 and R2, verify its properties.
• Repeat for the changed order of R1 and R2.
• Compute the logarithm matrix S1 of R1, and exponent of the result

  >>  S1=logm( R1 )
  >>  expm(S1)
  

  verify the properties of S1.

• Check transformations between a number om1 and a skew-symmetric matrix S1

  >>  om1=vex(S1)
  >>  S1b=skew(om1)
  

  What is a relation between om1 and theta? Does it hold for other angles?

• Repeat the last step for the product of two matrices for theta1 and theta2

  >>  R_pr=R1*R2
  >>  S_pr=logm(R_pr)
  >>  om_pr=vex(S_pr)
  

  What is a releation between om_pr, theta1 and theta2?

• Questions:
  1. What is the form of a rotation matrix on a plane, to what group it belongs, what are the properties of the elements of the group?
  2. What is the form of a logarithm of rotation matrix? To what group it belongs, what are its properties?
  3. What are properties of a product of two rotations?

• Define homogenous transformations for rotations and translations for two different angles and vectors

  >> Tt1=transl2(1, 2)
  >> Tr1=trot2(30, 'deg')
  >> Tr2=trot2(pi/6)
  

  Observe the structure of matrices, then plot the examples

  >> plotvol([0 5 0 5]);
  >> trplot2(Tt1,'frame', '1', 'color', 'r')
  

• Calculate compounds of transformations (two rotations, two translations, mixed rotation and translation in various order). Observe the properties of results.
• Check commutativity of compound (note: use also transformations with both angle and shift non-zero). Plot the results.
• Define a point with chosen coordinates P and homogenous transformation T1

  >> P=[3;1.5];
  >> plot_point(P, 'label', 'P', 'solid', 'ko');
  >> Ph=e2h(P)
  >> h2e(Ph)
  

  Find a point P1 having coordinates P in coordinate system defined by T1. Calculate its coordinates in original system and plot it.

• Find a position of P in coordinate system defined by T1.
• Questions:
  1. What is the form of a homogenous transformations and to what group it belongs?
  2. What are properties of compounds of transformations?
  3. How to transform coordinates of a point between an original and transformed coordinate frames (and back)?

• Define rotations around main axes (rotx, roty, rotz) and visualize them

  >> Rx = rotx(pi/6)
  >> trplot(Rx)
  

  check the properties as in task 1.

• Follow the steps of the task 1 in 3D case.
• Questions:
  1. What are similarities and differences between planar and spatial case?

• Choose 3 angles φ, θ, ψ representing Euler angles ZYZ. Calculate basic rotations around respective axes and a resulting compound rotation. Compare the result with the result of package function.

  >> R = eul2r(phi, theta, psi)
  

• For a rotation matrix calculate corresponding Euler angles.

  >> tr2eul(R)
  

  Observe the result.

• Repeat the previous steps for angles φ=0.1 , θ=-0.2, ψ=0.3.
• Follow the same steps in RPY representation

  >> rpy2r(0.1, 0.2, 0.3)
  >> tr2rpy(R)
  

• Questions:
  1. What is the reason of the result in step 3?
  2. Which 3 angles for RPY representation have the same property?
• Note: a tripleangle application may be used a help to understand the 3 angle representations

• Choose a rotation matrix R2 (with 2 or 3 non-zero parameters in Euler representation), let R1 be identity.
• Calculate axis-angle representation of R2 (tr2angvec)
• Choose another axis and angle of rotation, calculate corresponding matrix R3 (angvec2r)
• For one of above cases plot initial and final orientation and the axis of rotation, add intermediate orientations (by using intermediate angles and calculations in loop)
• (*) Extend the previous case by using translations to obtain a spiral motion along the axis, with a center of coordinate frame in a distance d from the axis of rotation
• To report:
  1. Include the plot and code illustrating the rotation around the axis

• Define and visualize exemplary transformations

  Tt=transl(1, 2, 3)
  Tr=eul2tr(0.3, 0.4, 0.5)
  trplot(Tt*Tr)
  

• Follow the steps from task 2
• Questions:
  1. What are similarities and differences between planar and spatial case?
  2. What is a structure of inverse matrix in 3D case?

• Define length parameters and angles of double pendulum
• Calculate points corresponding to the center of rotation and the end-points of each link using homogenous transformations.
• Plot the double pendulum and coordinate systems associated with each link
• Define initial and final angles, divide pendulum motion time to k periods, plot intermediate poses
• (*) use tranimate() or <object>.animate to show animated motion
• (**) use ode solver to calculate a solution of the pendulum motion, animate the result
• To report:
  1. Include code, parameters and final illustration of the example