Fаts, оils, аnd wаxes are best classified as which biоlоgical macromolecule?
Which оf the fоllоwing is а primаry goаl of the school externship experience?
Prоblem 1 - True/Fаlse (4 pts eаch; 40 pts) (а) There exists a rоtatiоnal matrix (Rin SO(3)) such that (R) is skew-symmetric. (b) For a given (Tin SE(3)), the adjoint transformation matrix (Ad_{T} in mathbb{R}^{6times 6}) is nonsingular. (c) For a given wrench (F_b) in ({b}), its equivalent wrench (F_c) in ({c}) satisfies (|F_b|=|F_c|). (d) It is possible to express forward kinematics for 4R-open chain manipulator in the following form: begin{equation*} T_{0b}(overline{theta})= e^{widehat{xi}_1theta_1}e^{widehat{xi}_2theta_2}Me^{widehat{xi}_3theta_3}e^{widehat{xi}_4theta_4},end{equation*} where (xi_i) are some screw axes and (theta_i) represent the relative joint angle for (i=1,2,3,4) and (M) is the home configuration. (e) Suppose you have a nonzero twist (xi_0inmathbb{R}^6). There exists a twist (xiinmathbb{R}^6) such that (Ad_{e^{widehat{xi}}}xi_0=xi_0) holds and (xi_0neq xi). (f) Suppose you have the inertia tensor of a single rigid body, relative to the body frame, (I_binmathbb{R}^{3times 3}). Then, (I_b) is a diagonal matrix. (g) Suppose that there is no external force acting on the system, then the Lagrangian (L(q(t),dot{q}(t))inmathbb{R}) of some mechanical systems with a generalized coordinate (qinmathbb{R}^n) is the same for all time (t). (h) If (f:mathbb{R}tomathbb{R}) is continuous, a nonlinear system (dot{x}=f(x)) with (x(0)=x_0 in mathbb{R}^n) has a unique solution for all time (t). (i) If the equilibrium (x=0) of (dot{x}=f(t,x)) is stable, then (lim_{ttoinfty}|x(t)|=0). (j) Lyapunov stability theorems do not require the explicit computation of solution trajectories ((x(t))). Problem 2 - Wrench (20 points) A bicopter is shown in the figure below. The spatial frame frame {s} is shown in black, the body frame frame {b} is shown in red, the left propeller frame frame {L} is shown in green, and the right propeller frame frame {R} is shown in yellow. Design parameters (ell, r), and (h) are shown with blue arrows. The transformation matrices (T_{bL}, T_{bR} in SE(3)) depend on the servo angles (delta_L, delta_R), respectively. As the propellers spin, they produce wrenches (mathcal{F}_L, mathcal{F}_R in mathbb{R}^6) which act on the frame {L} and frame {R} frames, respectively. (a) (10 points) Given (mathcal{F}_L, mathcal{F}_R in mathbb{R}^6) and (T_{bL}, T_{bR} in SE(3)), write an expression for (mathcal{F}_b), the total equivalent body wrench acting on the frame {b} frame, expressed in the frame {b} frame, using adjoint transformations. (b) (10 points) It can be shown that the equivalent body wrench (mathcal{F}_b) can be written as begin{equation*} mathcal{F}_b = begin{bmatrix} 0 & alpha & 0 & ell \ h & 0 & 0 & 0 \ 0 & -ell & 0 & alpha \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 end{bmatrix} begin{bmatrix} f_L sin delta_L + f_R sin delta_R \ f_L sin delta_L - f_R sin delta_R \ f_L cos delta_L + f_R cos delta_R \ f_L cos delta_L - f_R cos delta_R end{bmatrix},end{equation*} where (alpha > 0) is some constant, and (f_L, f_R > 0) are the thrusts of the propellers. What is the y-axis body torque when (h=0)? Why would this be problematic for controlling the bicopter's orientation, (R_{sb})? Problem 3 - Forward/Velocity Kinematics (30 points) A spatial RRPRPR open chain is shown in the figure below at its home configuration, with fixed frame ({s}) and end-effector frame ({b}). (a) (15 points) The forward kinematics can be expressed in the form$$T_{sb} = e^{hat{S}_1 theta_1}e^{hat{S}_2 theta_2}cdots e^{hat{S}_6 theta_6}M,$$where (Min SE(3)) is the home configuration and (S_1,ldots,S_6in mathbb{R}^6) are screw axes. Provide all entries of (M) and (S_1,ldots,S_6). (b) (15 points) Assume the open chain is at the home configuration.Find the end-effector spatial twist (mathcal{V}_s) when the joint velocity is $$dot{theta}=[1,1,1,1,1,1]^top.$$Also, express the linear velocity of the ({b})-frame origin with respect to the ({s})-frame. Problem 4 - Manipulability (30 points) A planar manipulator with a cam attached to joint (theta_1) is shown in the figure below. As (theta_1) rotates, the link in contact with the cam translates linearly along the (y)-axis. The shape of the cam is also described in the figure with (r(theta)=3R+Rcos theta). From the figure, we see thatbegin{align*} x &= Lcostheta_2 + x_0\ y &= 3R + Rcostheta_1 + Lsintheta_2 + y_0,end{align*}where (x_0) and (y_0) are constants depending on the choice of the origin. (a) 10 points) Derive the Jacobian (Jinmathbb{R}^{2times2}) relating the joint rates ([dot{theta}_1, dot{theta}_2]^top) to the tip velocity ([dot{x},dot{y}]^top). (b) (10 points) For the Jacobian (J) derived in (a), find all kinematic singularities. (c) (10 points) One scalar measure of a robot's manipulability is given by$$mu = sqrt{det(JJ^top)},$$which is the volume of the manipulability ellipsoide. Assuming (L=R=1) for the given manipulator, find the configuration ((theta_1,theta_2)) that maximizes (mu). Problem 5 - Energy (20 points) Suppose we have 3R open-chain manipulator. The inertial frame ({s}), body frames ({1}), ({2}), and ({3}) are defined as in the figure below (left). Suppose that the home configuration is also given as in the figure (right). Defining the generalized coordinate (q:=[q_1,q_2,q_3]^top), we havebegin{align*} T_{s1}(q) &= e^{hat{S}_1q_1}N_1, \ T_{s2}(q) &= e^{hat{S}_1q_1}e^{hat{S}_2q_2}N_2, \ T_{s3}(q) &= e^{hat{S}_1q_1}e^{hat{S}_2q_2}e^{hat{S}_3q_3}N_3end{align*}from the forward kinematics, where (S_i) are some screw axes and (N_i) are the home configurations of each frame ({i}) for (i=1,2,3). For obtaining the total kinetic energy, we can first compute the body velocity of ({3})-frame as$$hat{mathcal{V}}_{b3}(q) = T_{s3}^{-1}(q) dot{T}_{s3}(q).$$Express the matrix (J_3(q) inmathbb{R}^{6times 3}) such that$$mathcal{V}_{b3}(q) = J_3(q)begin{bmatrix} dot{q}_1\ dot{q}_2\ dot{q}_3end{bmatrix}$$by using adjoint transformations, (S_i), (N_i), and (q_i), (i=1,2,3). Show your work. Problem 6 - Dynamics (20 points) Suppose that a flapping-wing vehicle (FWV) is carrying a flower with a stick, and the connection between FWV, stick, and flower are all rigid. Therefore, it can be viewed as a single rigid body. Assume that the stick is massless, and we know the inertial properties of FWV, given as ((m_b, I_b)) where (m_b) is the total mass of FWV, and (I_b) is the inertial tensor of ({b})-frame attached to the center of mass (p_{sb}inmathbb{R}^3). Likewise, the inertial properties of the flower are given as ((m_c, I_c)), where (m_c) is the total mass of the flower, and (I_c) is the inertial tensor of ({c})-frame attached to the center of mass (p_{sc}inmathbb{R}^3). In summary, we have begin{equation*} T_{sb} = begin{bmatrix} R_{sb} & p_{sb} \ 0_{1times 3} & 1 end{bmatrix}, quad M_b = begin{bmatrix} I_b & 0_{3times 3} \ 0_{3times 3} & m_bI_{3times 3} end{bmatrix}, qquad T_{sc} = begin{bmatrix} R_{sc} & p_{sc} \ 0_{1times 3} & 1 end{bmatrix}, quad M_c = begin{bmatrix} I_c & 0_{3times 3} \ 0_{3times 3} & m_cI_{3times 3} end{bmatrix}.end{equation*} Assume that the total center of mass of the FWV carrying the flower is located at (p_{sa}inmathbb{R}^3), which is the red dot in the figure, and we attach ({a})-frame to this center of mass such that begin{equation*} T_{sa} = begin{bmatrix} R_{sa} & p_{sa} \ 0_{1times 3} & 1 end{bmatrix}.end{equation*} We want to find the expression of the generalized mass matrix (M_ainmathbb{R}^{6times 6}) such that the total kinetic energy is represented as begin{equation*} mathcal{T} = frac{1}{2}mathcal{V}_a^top M_amathcal{V}_a,end{equation*} where (mathcal{V}_a) is the body twist of ({a})-frame. Express the two matrices (M_{11}, M_{22} in mathbb{R}^{3times 3}) such that begin{equation*} M_a = begin{bmatrix} M_{11} & star \ star & M_{22} end{bmatrix}end{equation*} by using (m_b,~m_c,~I_b,~I_c,~R_{ba},~R_{ca},~p_{ba},~p_{ca}). Show your work. 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: 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: Final Exam 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.