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Uses аnd Grаtificаtiоns theоry is an extensiоn of Maslow's Hierarchy of Needs, and claims that people's use of media is goal-directed.
An аttribute thаt determines which subtype shоuld be used is а(n) ________.
Prоblem 1 - Cоnfigurаtiоn Spаce (40 pts) (а) (15 points) Suppose we have a walking robot modeled in the planar space as shown in Figure 1. The robot is configured with five rigid bodies, which are connected by two revolute joints (left hip joint and right hip joint) and two revolute joints (left knee and right knee). Suppose the robot stands with one leg and swings the other to walk forward. If we assume that the point foot is fixed on the ground but allowed to rotate (just like our ankle), then we can model it as another revolute joint, as shown in the figure. Figure 1: Picture from "Hybrid Zero Dynamics of Planar Biped Walkers", IEEE TAC, VOL. 48, NO. 1, 2003. How many degrees of freedom are there during this one-leg swinging phase? Use Grubler's formula to find the answer. You should provide answers for (N), (J), (m), (f), and DoF. (b) (15 points) We want to model the motions of a cart moving along an elliptical rail, where the major axis of the rail is (4) m and the minor axis of the rail is (2) m as shown in Figure 2. The elliptical rail is placed on the (XY)-plane, and so there is no elevation in (z_s)-axis during the motion. In addition, the cart doesn't roll along the axis of the rail and it always stays upright. Therefore, we can define an implicit coordinate for the configuration of the cart as (q(t):=[x(t), y(t), z(t), theta(t)]^top) observed in the fixed ({s})-frame, where (x(t), y(t), z(t)) are the ((X-Y-Z)) position of the center of mass of the cart. The (theta(t)) is representing the angle between the tangential vector of the rail at time (t), denoted as (T(t) = [ costheta(t), sintheta(t)]^topinmathbb{R}^2) in (XY)-plane, and the fixed (x_s)-axis. See the top-down view in the figure below. Let (z(t)= 1) m for all time (tgeq0). Figure 2: A cart on an elliptical rail. Therefore, we can write three holonomic constraints, where two constraints are already given in this problem. The first one is for the constant (z(t)) height and the second one is to ensure the ((x(t), y(t))) position stays on the elliptical rail as follows:begin{align} z(t) &= 1, hspace{1 cm} (1)\ frac{x(t)^2}{4}+y(t)^2&=1 hspace{1.2cm} (2)end{align}for all (tgeq0). The last holonomic constraint can be written by using the vector (T(t)) parmeterized by (theta(t)), that must be tangential to the rail. Observe that for every ((x(t), y(t))), we can write the vector (N(t) = [x(t),4y(t)]^top) orthogonal to the vector (T(t)) as shown in the figure. Therefore, we can construct the third holonomic constraint by (langle N(t), T(t)rangle =0) as follows: begin{equation}x(t)costheta(t)+4y(t)sintheta(t)=0. hspace{1cm} (3)end{equation}Lastly, one velocity constraint can be constructed to make sure (T(t)) is aligned with ((dot{x}(t), dot{y}(t))) as follows:begin{equation} - dot{x}(t) sin theta(t) +dot{y}(t) costheta(t)=0. hspace{1cm} (4)end{equation} Using the above four constraints (1)-(4), formulate the Pfaffian constraint as (A(q)dot{q}=0), where (A(q)inmathbb{R}^{4times 4}), and (dot{q}=[dot{x},dot{y},dot{z},dot{theta}]^top). Show your work for partial credit. (c) (10 points) A cart on a rail will be of one degree of freedom. This implies that three out of four constraints are independent. Show that the rank of (A(q)) in (b) is less than (4) when ((q,dot{q})) satisfies (1)-(4). Problem 2 - Rotational Motion (40 points) (a) (10 points) Let the rigid body have an initial orientation (R_{sb}). The following procedure describes how to express the body's orientation in terms of the ZYZ Euler representation in 3D. First, rotate the robot by an angle (alpha) about the body (z)-axis, then rotate the robot by an angle (beta) about the new body (y)-axis. Finally, rotate the robot by an angle (gamma) about the new body (z)-axis. Let (R_{sb^{ZYZ}}) denote the orientation of the resulting ({b^{ZYZ}})-frame, expressed in ({s})-frame. The following rotational matrices represent the rotations about (z)-, (y)-, and (z)-axes by angles (alpha) (beta) and (gamma) respectively: begin{equation*} Rot(z, alpha) = begin{bmatrix} cos{alpha} & -sin{alpha} & 0 \ sin{alpha} & cos{alpha} & 0 \ 0 & 0 & 1 end{bmatrix}, Rot(y, beta) = begin{bmatrix} cos{beta} & 0& sin{beta} \ 0 & 1 & 0 \ -sin{beta} & 0 & cos{beta} end{bmatrix}, Rot(z, gamma) = begin{bmatrix} cos{gamma} & -sin{gamma} & 0 \ sin{gamma} & cos{gamma} & 0 \ 0 & 0 & 1 end{bmatrix}.end{equation*} Express the (R_{sb^{ZYZ}}) in a matrix multiplication form using (R_{sb}, Rot(z, alpha), Rot(y, beta), Rot(z, gamma) ). You don't need to compute the matrix multiplication. (b) (10 points) Suppose that ((omega, theta)) is an exponential coordinate of some rotational matrix (Rin SO(3)) where begin{equation*} omega = frac{1}{sqrt{3}}begin{bmatrix} -1 \ 1\ 1 end{bmatrix}, qquad theta=frac{pi}{2}.end{equation*} Find a unit vector (xinmathbb{R}^3) such that begin{equation*} (I_3-R)x = 0_{3times 1},end{equation*} and explain the physical relation between the vector (x) and the rotation transformation (R). (Here, (I_3) denotes the (3)-dimensional identity matrix and (0_{3times 1} = [0,0,0]^top).) (c) (20 points) Suppose that a flapping-wing vehicle robot is flying in the motion capture system (MCS). When (t=t_1) the robot's orientation, shown in Figure~3 as ({b})-frame, is captured in ({s})-frame. Therefore, the orientation can be represented by (R_{sb}). Two seconds after, (t_2=t_1+2) the MCS captured the updated robot's orientation, shown in Figure 3 as ({b^prime})-frame. This orientation can be represented by (R_{sb^prime}). Figure 3: Flapping-wing vehicles in the air. Suppose that (R_{sb}) and (R_{sb^prime}) are given as follows: begin{equation*} R_{sb}=begin{bmatrix} 1 & 0& 0\ 0 & 0& -1\ 0 & 1& 0 end{bmatrix}, qquad R_{sb^prime}=begin{bmatrix} 0 & -1& 0\ 1 & 0& 0\ 0 & 0& 1 end{bmatrix}. end{equation*} (This may not match Figure 3, so ignore the axes and geometry shown there.) (i) What is the orientation of ({b^prime})-frame observed in ({b})-frame? Provide all (9) entries of the (3times 3) matrix as numerical values. (ii) In this question, we calculate how much the robot rotated from (t= t_1) to (t=t_2). Consider that the robot was rotated from ({b})-frame to ({b^prime})-frame along a certain axis represented in ({b})-frame. What is the unit axis of rotation (omega) and the angle (thetain [-pi, pi)) observed in ({b})-frame? (iii) What is a good approximation for the body angular velocity [rad/second]? (Hint: Think about the physical meaning of the body angular velocity (omega_b) we learned in class.) Problem 3 - Rigid Body Representation (45 points) In this problem, we consider the full rigid body coordinates in (SE(3)). Suppose we want to control our flapping-wing vehicle robot to land on a flower petal. Figure 4: Flapping-wing vehicle robots in the air with the full coordinate in (SE(3)). We know the petal's position and orientation in ({s})-frame and the coordinates of ({b})-frame expressed in ({s})-frame as follows: begin{equation*} p_{sc} = begin{bmatrix} 2\ 1\ 3 end{bmatrix},~~ R_{sc} = begin{bmatrix} 0 & 0 & 1\ 0 & 1 & 0\ -1& 0 & 0 end{bmatrix},~~ p_{sb} = begin{bmatrix} 1\ 1\ 5 end{bmatrix}, ~~R_{sb}=begin{bmatrix} 1 & 0& 0\ 0 & 0& -1\ 0 & 1& 0 end{bmatrix}.end{equation*} (This may not match Figure 4, so ignore the axes and geometry shown there.) (a) (10 points) Write down the homogeneous representation of the ({s})-frame, expressed in the ({b})-frame. Provide all (16) entries of the (4times 4) matrix as numerical values. (b) (20 points) Now, consider the flapping-wing vehicle robot flying to the petal by following some screw motion to arrive at the desired petal with the exact orientation matching with ({c})-frame. In this question, we want to find the rigid transformation (T:SE(3)to SE(3)) in the body ({b})-frame such that begin{equation*} T_{sc} = T_{sb}T.end{equation*} (i) Find the transformation matrix (T). Then, decompose it with pure translation (T_pin SE(3)) and pure rotation (T_Rin SE(3)) matrices in a matrix multiplication form. Provide all (16) entries of all (4times 4) matrices as numerical values. (ii) Draw ({b})-frame and ({c})-frame in the following figure representing ({s})-frame and explain how the origin of the ({b})-frame has been moved to the origin of the ({c})-frame by using the pure translation matrix (T_p) obtained in (i). (c) (15 points) Let us consider the coordinates of ({b'})-frame expressed in ({s})-frame as follows: begin{align*} p_{sb'} = begin{bmatrix} 0 \ sqrt{2} \ 0 end{bmatrix}, qquad R_{sb'} = begin{bmatrix} 1 &0&0\0&0&-1\0&1&0 end{bmatrix}.end{align*} (Here, (I_3) denotes the (3)-dimensional identity matrix.) Suppose that a transformation matrix is given as $$T' = begin{bmatrix} 0 & 1 & 0 & pi\ 1 & 0 & 0 & pi \ 0 & 0 & -1 & 2sqrt{2} \ 0 & 0 & 0 & 1end{bmatrix},$$ which is also represented by (T'=e^{hat{S}'theta'}), where begin{align*} S' = begin{bmatrix} omega' \ v' end{bmatrix} = begin{bmatrix} frac{1}{sqrt{2}} \ frac{1}{sqrt{2}} \ 0 \ 0 \ 2 \ 0 end{bmatrix}inmathbb{R}^6, qquad omega',v'inmathbb{R}^3, qquad theta' = pi inmathbb{R}.end{align*} are expressed in ({b'})-frame. Find the screw pitch (h'inmathbb{R}) and the point (q_{b'}inmathbb{R}^3) attached to the axis of the screw motion represented in ({b'})-frame. Then, compute (T_{sb''}= T_{sb'}T') and draw ({b'})-frame, ({b''})-frame, and the screw axis in the following figure representing ({s})-frame. Explain the physical meaning of (omega' in mathbb{R}^3) and (h'theta'inmathbb{R}) based on the figure. Congratulations, you are almost done with this exam. . DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Midterm Exam Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to Gradescope to submit your work: Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.