Robotics I Midterm classroom test November 24, 2017

Transcription

1 xercise [8 points] Robotics I Midterm classroom test November, 7 Consider the -dof (RPR) planar robot in ig., where the joint coordinates r = ( r r r have been defined in a free, arbitrary way, with reference to a base frame R b. y e N L P e x e r r y b = x r x b = y igure : RPR planar robot. etermine position and orientation of the end-effector frame R e in terms of the coordinates r. ssign the enavit-hartenberg (H) frames and define the joint coordinates q = ( q q q according to the standard H convention, starting from the reference frame R. Provide the H table, the expression of the homogeneous transformation matrices between the successive frames that have been assigned. etermine position and orientation of the end-effector frame R e in terms of the coordinates q. ind the transformation q = f(r) between the two sets of coordinates so as to associate the same kinematic configurations of the robot. oes this mapping have singularities? xercise [ points] encoder C motor! m spur gear: r =.8, r =. [m] max motor torque:! m =.8 [Nm] bevel gear: r = r =. [m]! L link inertia: J L =. [kg m ] link: L =. [m] P ee igure : C servomotor that drives a robot link through transmissions. With reference to the servo-drive sketched in ig. and the data therein, we need to measure the position of the tip point P ee of the link with a resolution of. [mm]. suitable incremental

2 R R encoder with quadrature detection is chosen to measure the angular position θ m of the C motor, which is capable of delivering a maximum torque τ m. How many pulses per turn should generated by the optical disk on each of the channels and B? How many bits should be used by the digital counter of the encoder? Neglecting dissipative effects, what should be the value of the motor inertia J m in order to maximize the angular acceleration of the link for a given motor torque? With this choice, what is the maximum linear acceleration achievable by the tip of the link? xercise [ points] MOL RX RXL Characteristics Consider the -dof robot Stäubli RX in ig.. In the extra sheet provided separately, the enavit-hartenberg (H) table of parameters is specified, in part numerically and in part symbolically. The two H frames and are already drawn on the manipulator (in two views). In the shown straight upward robot configuration, the first and last joint variables take the values q = q =. raw directly on the extra sheet the remaining H frames, according to the H table. Provide all parameters labeled in red in the table, i.e., the missing numerical values of the RX dimensions constant parameters and of the joint variables q to q when the robot is in the straight upward configuration. [Please, make clean drawings and return the sheet with your name written on it.] kg 8 kg kg kg and ) 7 mm mm f freedom 8 ±, mm ±, mm ± ± ± 7, ± 7, ± ± ± 7 ± 7 + /- + /- Work envelope ± 7 () ± 7 () h between axis and (R. M) mm 9 mm between axis and (R. m) mm mm between axis and (R. m) mm mm between axis and (R. b) mm 9 mm /s /s Maximum speed Maximum inertias orearm connections /s /s B C i! i a i d i " i!/ a > d > q = a > q!/ q!/ d > q!/ q d > q = /s /s /s igure : The /s Stäubli RX robot manipulator and the extra sheet (provided separately). /s /s 87 /s 87 /s ad gravity center, m/s, m/s solenoid valves /-way (compressed air) monostable direct line between the base and the forearm female 9-contact socket (7 twisted pairs including shielded, power contacts) - ISO - ist) d N N 9 ntroller according to standard - o standard directive Installation environment xercise [ points] () Version H (Humid nvironment) : designed for use in humid and oxidizing environments. The arm components are painted individually, providing additional arm protection against oxidation and corrosion. actory installation only and required with Pressurization kit. etermine, IP (*7) if possible, a unit vector r and an angle θ < such that the axis-angle rotation matrix R(r, θ) provides CS8C the orientation of the frame R B with respect to the frame R. + C to + C % à 9% max. non-condensing loor/ceiling ) B C () Under special conditions, consult us. () Software configurable up to ±8... Staubli RX ssignment of frames according to the given H table Name:!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"" z " Wrist z " x " x " view from the right side y " y " z " z " view from the back side Robotics, Midterm classroom test, November, 7 The orientations of two right-handed frames R kg.m,8 kg.m and R B with respect to a third right-handed frame kg.m R (all,7 having kg.m the same origin) are specified, respectively, by the rotation matrices 8 kg kg () Under special conditions, consult us. () Software configurable up to ±8. ll axis 8 () Pressurization kit : necessary for use in an environment with high dust levels or with substantial liquid splashing. This kit generates positive pressure in the arm. R = actory installation only and required with 8 8 and R B = Pressurization kit.. 8 [8 minutes, open books but no computer or smartphone] RXL dimensions Mount nt) version) () Market specific versions s cleanliness - ISO - () Pressurization kit : necessary for use in an environment with high dust levels or with substantial liquid splashing. This kit generates positive pressure in the arm. actory installation only and required with Pressurization kit. o standard directive actory Mutual or ccsus (US) () Version H (Humid nvironment) : designed for use in humid and oxidizing environments. The arm components are painted individually, providing additional arm protection against

3 Solution of Midterm Test November, 7 xercise The direct kinematics in terms of the coordinates r = ( r r r is easily computed as b p e (r) = (r + N) cos r + L cos r (r + N) sin r + L sin r b R e (r) = cos r sin r sin r cos r. Indeed, if these quantities have to be expressed with respect to the (H) frame indicated in ig., we need to introduce a constant rotation. Since the two frames have the same origin, we obtain R b = p e (r) = R b b p e (r) = R e (r) = R b b R e (r) = (r + N) sin r + L sin r (r + N) cos r L cos r sin r cos r cos r sin r. Keeping into account the already specified assignment of frame in ig., a feasible assignment of H frames for the RPR robot is shown in ig., with associated parameters given in Tab.. x y = y e x y q q L P e x = x e z q x igure : ssignment of H frames for the RPR planar robot. i α i a i d i θ i π/ q π/ q L q Table : Parameters associated to the H frames in ig..

4 The H homogenous transformations take the following expressions: cos q sin q sin q cos q (q ) = (q ) = q cos q sin q L cos q sin q cos q L sin q (q ) =. The direct kinematics in terms of the coordinates q = ( q q q is then T e (q) = (q ) (q ) (q ) = ( R e (q) p e (q) ), with p e (q) = q sin q + L cos(q + q ) q cos q + L sin(q + q ) R e (q) = cos(q + q ) sin(q + q ) sin(q + q ) cos(q + q ). The transformation that matches the robot configurations using the different sets of coordinates is q = f(r) = r r + N r r π r = f (q) = q N q + q + π which is invertible and without singularities. It is easy to check that, e.g., q p e (q) q=f(r) = p e (r), R e (q) q=f(r) = R e (r). xercise The requested resolution = [m] on the linear motion at the tip of the link of length L =. [m] needs to be transformed into an angular one δ = /L =.88 [rad] at the base of the link and then, via the two transmission gears with reduction ratios N bevel = r /r = and N spur = r /r =./.8 =.7 respectively, into an angular resolution on the motor axis δ m = N spur N bevel L = 7.9 [rad] = (. ). Taking into account the factor introduced by the quadrature electronics, the number of pulses per turn N ppt of the optical disk should be at least π N ppt = = 978. δ m ccordingly, the digital counter should have at least a number of bits N bit = log 978 =.

5 R e Being the link inertia (around its axis of rotation) J L =. [kg m ] and the reduction ratio of the transmissions N = N spur N bevel =.7, the optimal value of the motor inertia according to the requested criterion is J m = J L N =.8 [kg m ]. ccordingly, with a maximum available motor torque τ m =.8 [Nm] on the motor axis, the maximum torque at the link base is τ L,max = N τ m =. [Nm]. Since N = J L /J m, the balance of torques on the link side provides RX dimensions τ L = J L θl + N(J m θm ) = (J L + J m N ) θ L = J L θl. The maximum tip acceleration a max of the link tip will then be a max = L θ L,max = L τ L,max =. [m].78 [rad s ] =.88 [m s ]. J L RXL xercise view from the right side z! view from the back side z! z! x! y! z! B C x! x! x, x! z! x! z! z! z! y! y! i! i a i d i " i!/ q = 8 q =!/!/ q =!/ z! x! x! y! z! z!!/ q =!!/ q = q = igure : The H frames and the complete table of parameters for the Stäubli RX robot manipulator. The numerical values of q = θ refer to the shown straight upward configuration. The assignment of enavit-hartenberg frames for the Stäubli robot RX according to the given table is shown in ig.. The numerical values of all symbolic parameters are reported, with the values of the joint variables q = θ when the robot is in the straight upward configuration. Wrist

6 xercise The relative orientation of frame R B with respect to frame R is expressed by the rotation matrix R B = R T R B = = whose elements will be denoted by R ij. Therefore, the equation R(r, θ) = R B should be solved for r and θ, using the inverse mapping of the axis-angle representation. Since sin θ = ± (R R ) + (R R ) + (R R ) = ±.999, () the problem at hand is regular, and two distinct solutions can be found depending on the choice of the + or sign in the expression of sin θ. rom cos θ = (R + R + R ) =.8, taking the sign in () will yield a solution angle θ <, as requested. Thus θ = TN {.999,.8} =.87 [rad] = 7.7 and r = sin θ R R R R = R R