DYNA http://dyna.medellin.unal.edu.co/

The dynamic model of a four control moment system Modelo dinámico de un sistema de control de par por cuatro giróscopos

Eugenio Yime-Rodríguez a, César Augusto Peña-Cortés b & William Mauricio Rojas-Contreras c

a Facultad de Ingenierías, Universidad Tecnológica de Bolívar, Colombia. [email protected] b Facultad de Ingenierías y Arquitecturas, Universidad de Pamplona, Colombia. [email protected] c Facultad de Ingenierías y Arquitecturas, Universidad de Pamplona, Colombia. [email protected]

Received: November 28th, 2012. Received in revised form: February 20th, 2014. Accepted: May 9th, 2014.

Abstract The dynamic model of a Four Control Moment Gyroscope (4-CMG) is traditionally obtained after computing the derivative of the equation. Although this approach leads to a simple dynamic model, new models have been introduced due to terms not taken into account during the computation of the angular momentum equation. In this paper, a new dynamic model for a 4-CMG based on the Newton-Euler algorithm, which is well accepted in Robotics, was developed. This new approach produces a complete dynamic model.

Keywords: dynamics, gyroscope, model; control.

Resumen El modelo dinámico de un sistema de control de par utilizando cuatro giróscopos (4-CMG) tradicionalmente se obtiene al calcular la derivada de la ecuación del momento angular total. Aunque este enfoque conduce a un modelo dinámico relativamente simple, recientemente se han introducido nuevos modelos debido a términos que no se han tenido en cuenta, o se desprecian, durante el cálculo de la ecuación del momento angular. En este artículo, se propone un nuevo modelo dinámico para un 4-CMG basado en el algoritmo de Newton-Euler, el cual es bien aceptado en robótica. Con este nuevo enfoque se logra obtener un modelo dinámico bastante completo.

Palabras clave: dinámica; giróscopo; modelo; control.

1. Introduction In this paper a kinematics comparison between a 4-CMG and a robotic arm is used to develop a new dynamic model. A Four Control Moment Gyroscope (4-CMG) is an angular The advantages of this approach are: use of a widely accepted momentum exchange device used on satellites [1-3], submarines methodology to compute a dynamic model, and more precise [4, 5], to control attitude. It is composed by four equations. arranged in a pyramidal form, see Fig. 1, with the amplification property being its principal advantage [6]. 2. Dynamic equations for A CMG Moreover, when used in satellites no fuel or gas propellant is required, because the motors use electricity to operate. Fig. 1 illustrates a CMG with four gyroscopes, each of The dynamic model of a 4-CMG is usually obtained by them composed of a flywheel and a gimbal. A coordinate differentiation of the angular momentum equation [7]. This frame 퐱i, 퐲i, 퐳i, is located at the center of the flywheel, which is done in [3, 4] and [8]. Probably the most exact dynamic serves as a reference for the motion of gimbals and flywheels. model using this approach is the developed by Ford and Hall, The flywheels turn at a constant speed, while the gimbals can [8]. rotate around 퐲i axis without any restriction. The angle of The first comparison between a robot arm and a 4-CMG rotation of a 푖-th gimbal is represented by 휃푖, with the zero was performed by Bedrossian et al. [9]; in this work an position being illustrated in the figure. The position of the analogy of velocities was considered to study the four gyroscopes is denoted by the vector 훉 = 푇 singularities on a 4-CMG. No further analogies with robot [휃1, 휃2, 휃3, 휃4] . The angle β is the pyramid’s skew arms were stated. angle.

© The authors; licensee Universidad Nacional de Colombia. DYNA 81 (184), pp. 41-47. June, 2014. Medellín. ISSN 0012-7353 Printed, ISSN 2346-2183 Online Yime-Rodríguez et al / DYNA 81 (184), pp. 41-47. June, 2014.

Figure 1. Control moment gyroscope with pyramidal array.

Fig. 2 illustrates the equivalent open kinematics chain for Figure 3. Gyroscope kinematics and Base body dynamics. a 4-CMG; there the circles represent rotational joints. Table 1. To compute the dynamic model, two steps are performed, Recursive Newton-Euler formulation. Forward kinematics. see Fig. 3. The first step is the Newton-Euler algorithm for Forward kinematics each gyroscope, this led to the reaction forces and moments Angular velocity propagation applied to the base body. The second step is the dynamic cos(휃i) sin(휃i) 0 i ( ) ( ) ( ) ( ) ( ) equation for the base body, where the reaction forces and 퐑i = [−cos 훼i sin 휃i cos 훼i cos 휃i sin 훼i ] sin(훼i) sin(휃i) −sin(훼i) cos(휃i) cos(훼i) moments exerted by each gyroscope, 퐌푖 and 퐅푖, are involved. i−1 푇 퐙i−1 = [0 0 1] The angular and linear velocity of the base body, 퐯0 and 훚0 Angular acceleration propagation ̈ ̇ respectively, plus the angular velocities of each joint, 훚푖,1 훚̇ i = 훚̇ i−1 + 퐳i−1휃i + 훚i−1 + 퐳i−1θi i훚̇ = i푹 i−1훚̇ + i−1퐳 휃̈ + i−1훚 + i−1퐳 θ̇ and 훚푖,2 are required in the Newton-Euler algorithm to i i−1( i−1 i−1 i i−1 i−1 i) perform the direct kinematics of the gyroscopes [10, 11]. Linear velocity propagation 퐯i = 퐯i−1 + 훚i × 퐫i Computation of the Newton Euler Equations for a serial i i i−1 i i 퐯i = 푹i−1 퐯i−1 + 훚i × 퐫i robot is also done using two steps, Tsai [12]. The first one is i 푇 퐫i = [ai di sin(훼i) di cos(훼i)] the kinematics calculus toward the extreme of the robot, as Linear acceleration propagation shown in Table 1. 퐯̇i = 퐯̇i−1 + 훚̇ i × 퐫i + 훚i × (훚i × 퐫i) i i i−1 i i i i i 퐯̇i = 푹i−1 퐯̇i−1 + 훚̇ i × 퐫i + 훚i × ( 훚i × 퐫i) Linear acceleration of the center of mass 퐯̇ci = 퐯̇i + 훚̇ i + 훚̇ i × 퐫ci + 훚i × (훚i × 퐫ci) i i i i i i i i 퐯̇ci = 퐯̇i + 훚̇ i + 훚̇ i × 퐫ci + 훚i × ( 훚i × 퐫ci) Acceleration of gravity i i i−1 퐠 = 퐑i−1 퐠

The second step is the dynamic calculus of backward computation, as can be seen in Table 2. Note, only rotational joints are considered in both tables.

Table 2. Recursive newton-euler formulation. Backward dynamics. Backward dynamics Inertial forces and moments ∗ 퐟i = −푚i퐯̇ci ∗ 퐧i = −퐈i훚̇ i − 훚i × (퐈i훚i) Force and torque balance equations about the center of mass ∗ 퐟i,i−1 = 퐟i+1,i − 푚i퐠 − 퐟i Figure 2. Equivalent kinematic chain of a 4-CMG. ∗ 퐧i,i−1 = 퐧i+1,i + (퐫i + 퐫ci) × 퐟i,i−1 − 퐫ci × 퐟i+1,i − 퐧i Torque in rotational joint i−1 푇 i−1 훕i = 퐧i,i−1 풛i−1

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The following assumptions have been made to simplify 푐푎푖 −푠푎푖푐훽 −푠푎푖푠훽 −푠푎푖푟 the equations of the Newton-Euler methodology: 푏 푠푎푖 푐푎푖푐훽 푐푎푖푠훽 푐푎푖푟 퐀0,푖 = [ ] (1) • Mass and inertia of the gimbals are approximately zero 0 −푠훽 푐훽 0 or negligible. 0 0 0 1 • The center of mass of the Flywheel is aligned with the gimbal axis. Where 푠푎푖, 푐푎푖, 푠훽, and 푐훽 stands for 푠푖푛(푎푖), cos (푎푖), • Velocity and acceleration of the base body are not equal sin (훽), and cos (훽) respectively; 푟 is the radius of the circle to zero. where the gyroscopes are located; 푎푖 is the angle of the turn The mass and inertia of the gimbal frame is neglected around axis 퐳0 to align 퐲0 with 퐲푖, and it has any of the values because the flywheel has the major contribution in the mass of {0, 휋/2, 휋, 2휋/3} radians. and inertia of the gyroscopes. The Denavit-Hantemberg parameters for one gyroscope according to Fig. 4, are shown in Table 3. In this table 퐿 = 2.1. Data of the links ‖퐫ퟏ,풊‖, where 퐫ퟏ,풊 is the vector from frame {0, 푖} to frame {1, 푖}. For the Newton-Euler approach, one gyroscope is composed of three links: base body, gimbal and flywheel. Table 3. Each of these links is joined by a rotational joint, Fig. 3. Denavit - Hartemberg parameters. Joint - i 훼 푎 푑 휃 Before computing the forward kinematics and backward i i i i 1 휋/2 0 L 휃i dynamics, each link must have a coordinate frame. Fig. 4 − 휋/2 illustrates the frames and vectors defined for each link. A new 2 0 0 0 휃i,2 frame, [퐱0,푖 퐲0,푖 퐳0,푖], is used to express the forces and moment exerted by the gyroscopes. This frame is fixed in the 1) Homogeneous Transformation Matrices: The DH base body. The other two frames are [퐱1,푖 퐲1,푖 퐳1,푖] and transformation matrices for each link can be also computed, [퐱2,푖 퐲2,푖 퐳2,푖] , with the former being the frame of the these are: gimbals link and the latter being the frame of the flywheel link. These two frames are located at the same point, the sin(휃푖) 0 −cos(휃푖) 0 center of the flywheel. The homogeneous matrix between the 0,푖 −cos(휃푖) 0 − sin(휃푖) 0 퐀1,푖 = [ ] (2) frames fixed in the base body, [퐱0 퐲0 퐳0] and 0 1 0 퐿

[퐱0,푖 퐲0,푖 퐳0,푖], is, 0 0 0 1

cos(휃 ) − sin(휃 ) 푖,2 푖,2 0 0 1,푖 sin(휃 ) cos(휃 ) 0 0 퐀2,푖 = 푖,2 푖,2 (3) 0 0 1 0 [ 0 0 0 1]

2) Position Vectors: The position of the frame {1, 푖} with 0,푖 2,푖 respect to {0, 푖} isdefined by vector 퐫푐0,푖. The vector 퐫2,푖 defines the position of {2, 푖} related to {1, 푖}. The mass centre 0,푖 1,푖 of each link is defined by vectors 퐫푐0,푖 and 퐫푐1,푖. These vectors have the following values,

1,푖 푇 퐫1,푖 = [0 0 퐿] (4) 2,푖 푇 퐫2,푖 = [0 0 0] (5) 0,푖 푇 퐫푐0,푖 = [0 0 0] (6) 1,푖 푇 퐫푐1,푖 = [0 0 0] (7)

0,푖 퐫푐0,푖 is zero because the mass and inertia of the gimbals are neglected.

3) Inertia and mass for links: For both links the values of mass and inertia are,

푚1,푖 = 0 (8) 푚2,푖 = 푚2 (9) Figure 4. Frames used in the Newton-Euler algorithm. 1,푖퐈 = ퟎ (10) 1,푖 2,푖퐈 = 푑푖푎푔([퐼 퐼 퐼 ]) (11) 2,푖 푥푥 푦푦 푧푧

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4) Base Body conditions: Different to a typical robot, the 2.3. Backward Dynamics base body of a 4-CMG is in motion, which allows it to have an angular and linear velocity as well as non-zero The following dynamics equations for one gyroscope are acceleration. obtained after applying the equations in table 2. 0,푖 0,푖 0,푖 0,푖 훚0,푖, 훚̇ 0,푖, 퐯0,푖, 퐯̇0,푖, not equal to zero. 1) Second Body: By using backward dynamics, the torque required by the flywheel motor can be computed as the 2.2. Forward Kinematics Inertial Forces and Moments

∗ The following equations are derived after using the 퐟2,푖 = −푚2퐯̇0,푖 − 푚2훚̇ 0,푖 × 풓1,푖 − 푚2훚0,푖 × 훚0,푖 × 퐫1,푖 (22) forward kinematics, table 1. ∗ ̇ 퐧2,푖 = −퐈2,푖훚̇ 0,푖 − 훚0,푖 × 퐈2,푖훚0,푖 − 휃푖I푦푦훚0,푖 × 풚1,푖 1) First link - gimbal axis: The first link has the following − 휃̇2,푖I푧푧훚0,푖 × 풛1,푖 − 휃̈푖I푦푦풚1,푖 (23) velocity and acceleration. − 휃̇푖휃̇2,푖I푧푧풙1,푖

Angular Velocity Where the following relations were used,

훚1,푖 = 훚0,푖 + 풛0,푖휃̇푖 (12) 퐈2,푖퐳0,푖 = 퐼푦푦퐲1,푖 (24) 퐈2,푖퐳1,푖 = 퐼푧푧퐳1,푖 (25) Angular Acceleration 퐈2,푖퐱1,푖 = 퐼푥푥퐱1,푖 (26)

훚̇ 1,푖 = 훚̇ 0,푖 + 풛0,푖휃̈푖 + 훚0,푖 × 풛0,푖휃̇푖 (13) Forces and Moments in the center of mass

퐟 = −푚 퐠 + 푚 퐯̇ + 푚 훚̇ × 풓 Linear Velocity 21,푖 2 2 0,푖 2 0,푖 1,푖 (27) + 푚2훚0,푖 × 훚0,푖 × 퐫1,푖 ̇ 퐯1,푖 = 퐯0,푖 + 훚0,푖 × 풓1,푖 (14) 퐧2,푖 = 퐈2,푖훚̇ 0,푖 + 훚0,푖 × 퐈2,푖훚0,푖 + 휃푖I푦푦훚0,푖 × 풚1,푖 + 휃̇2,푖I푧푧훚0,푖 × 풛1,푖 + 휃̈푖I푦푦풚1,푖 (28) Because 풛0,푖 and 풓1,푖 are parallel. + 휃̇푖휃̇2,푖I푧푧풙1,푖 Linear Acceleration Torque in joint 퐯̇1,푖 = 퐯̇0,푖 + 훚̇ 0,푖 × 풓1,푖 + 훚̇ 0,푖 × 훚̇ 0,푖 × 풓1,푖 (15) 푇 푇 훕2,푖 = 퐳1,푖퐈2,푖훚̇ 0,푖 + 퐳1,푖훚0,푖 × 퐈2,푖훚0,푖 ̇ 푇 (29) Acceleration of the center of mass + 휃푖I푦푦퐳1,푖훚0,푖 × 풚1,푖

퐯̇푐1,푖 = 퐯̇1,푖 (16) 2) First Body: In these steps the torque required by gimbal motor and the reaction moments and forces in the base body 2) Second Link - flywheel axis: Preforming the same are computed. Inertial Forces and Moments steps, as the first link, the results are, ∗ 퐟1 = 0 (30) ∗ Angular Velocity 퐧1 = 0 (31)

훚2,푖 = 훚0,푖 + 풛0,푖휃̇푖 + 풛1,푖휃̇2,푖 (17) Forces and Moments in the center of mass

Angular Acceleration 퐟 = −푚 퐠 + 푚 퐯̇ + 푚 훚̇ × 풓 10,푖 2 2 0,푖 2 0,푖 1,푖 (32) + 푚2훚0,푖 × 훚0,푖 × 퐫1,푖 훚̇ 2,푖 = 훚̇ 0,푖 + 풛0,푖휃̈푖 + 훚0,푖 × 풛0,푖휃̇푖 + 훚0,푖 × 풛1,푖휃̇2,푖 (18) ̇ + 휃̇푖휃̇2,푖퐱1,푖 퐧10,푖 = 퐈2,푖훚̇ 0,푖 + 훚0,푖 × 퐈2,푖훚0,푖 + 휃푖I푦푦훚0,푖 × 풚1,푖 + 휃̇2,푖I푧푧훚0,푖 × 풛1,푖 + 휃̈푖I푦푦풚1,푖 Linear Velocity + 휃̇푖휃̇2,푖I푧푧풙1,푖 − 푚2풓1,푖 × 품 (33) 퐯2,푖 = 퐯0,푖 + 훚0,푖 × 풓1,푖 (19) + 푚2풓1,푖 × 퐯̇0,푖 + 푚2풓1,푖 × 훚̇ 0,푖 × 풓1,푖 + 푚2풓1,푖 × 훚0,푖 × 훚0,푖 × 풓1,푖 Linear Acceleration 퐯̇2,푖 = 퐯̇0,푖 + 훚̇ 0,푖 × 풓1,푖 + 훚̇ 0,푖 × 훚̇ 0,푖 × 풓1,푖 (20) Torque in Joint

푇 푇 Acceleration of the center of mass 훕1,푖 = 퐳0,푖퐈2,푖훚̇ 0,푖 + 퐳0,푖훚0,푖 × 퐈2,푖훚0,푖 ̇ 푇 ̈ (34) 퐯̇푐2,푖 = 퐯̇2,푖 (21) + 휃2I푧푧퐳0 훚0,푖 × 풛1,푖 + 휃푖I푦푦

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2.4. Dynamic Equation for Base Body ퟒ ′ ∑ 푚2퐫푖 × 퐯̇푏 = 0 (52) The total force and moment exerted on the base body, is 풊=ퟎ the sum of the force and torque for each gyroscope’s 풂 × 풃 × 풃 × 풂 = −풃 × 풂 × 풂 × 풃 (53) equation, (32) and (33). The obtained result is, 4 ퟒ

퐅푒푥푡 + 푚0품 = 푚0퐯̇푏 + ∑ 퐟10,푖 (35) ∑(푛10,푖 + 퐫풊 × 퐟10,푖) 푖=1 풊=ퟎ 4 ퟒ 훕 = 퐈 훚̇ + 훚 × 퐈 훚 + ∑(퐧 + 퐫 × 퐟 ) (36) ′ ′ 푒푥푡 푏 푏 푏 푏 푏 10,푖 푖 10,푖 = ∑(퐈2,푖흎̇ 푏 − 푚2퐫푖 × 퐫푖 × 흎̇ 푏 (54) 푖=1 풊=ퟎ ′ ′ + 흎푏 × 퐈2,푖흎푏 − 푚2흎푏 × 퐫푖 × 퐫푖 If 퐫푖, 퐯0,푖, 퐯̇0,푖, 훚0,푖 and 훚̇ 0,푖 are defined by the following × 흎푏 + 휃̇푖I푦푦흎푏 × 풚1,푖 + 휃̇2,푖I푧푧흎푏 expressions, × 풛1,푖 + 휃̈푖I푦푦풚1,푖 + 휃̇푖휃̇2,푖I푧푧풙1,푖) 푇 퐫1 = [0, 푟, 0] (37) 푇 퐫ퟐ = [−푟, 0, 0] (38) It is a common practice to represent the torque’s equation 푇 퐫3 = [0, −푟, 0] (39) in the base body coordinate frame. In this frame, the 푇 퐫4 = [푟, 0, 0] (40) following matrices are defined, 흎0 = 흎푏 (41) 흎̇ 0 = 흎̇ 푏 (42) 푏 푏 푏 퐗1 = [ 푥1,1 … 푥1,4] (55) 퐯0,푖 = 퐯푏 + 흎푏 × 퐫푖 (43) 푏 푏 푏 퐘1 = [ 푦1,1 … 푦1,4] (56) 퐯̇0,푖 = 퐯̇푏 + 흎̇ 푏 × 퐫푖 + 흎푏 × 흎푏 × 퐫푖 (44) ̇ 푇 휽 = [휃1̇ … 휃̇4] (57) ̈ 푇 휽 = [휃1̈ … 휃̈4] (58) Then eq. (32)-(33) are expressed in terms of base body ퟒ variables, 푏 푏 푏 풊 푏 푇 푏 ′ 푏 ′ 퐈푡 = 퐈푏 + ∑( 퐑푖 퐈푖 퐑푖 − 푚2 퐫̃푖 퐫̃푖 ) (59) 풊=ퟎ ퟒ 퐟10,푖 = −푚2퐠 + 푚2퐯̇푏 + 푚2훚̇ 푏 × 풓푖 푏 푏 풊 푏 푇 푏 ′ 푏 ′ + 푚2훚푏 × 훚푏 × 퐫푖 + 푚2훚̇ 푏 × 퐫1,푖 (45) 퐃푖 = ∑( 퐑푖 퐈푖 퐑푖 − 푚2 퐫̃푖 퐫̃푖 ) (60) + 푚2훚푏 × 훚푏 × 퐫1,푖 풊=ퟎ 푏퐈 = 푏퐈 + 푏퐃 (61) 퐧10,푖 = 퐈2,푖훚̇ 푏 + 훚푏 × 퐈2,푖훚푏 + 휃̇푖I푦푦훚푏 × 풚1,푖 푡 푏 푖

+ 휃̇2,푖I푧푧훚푏 × 풛1,푖 + 휃̈푖I푦푦풚1,푖 Where 퐚̃ is the matrix equivalent of the cross product, 퐚 × + 휃̇푖휃̇2,푖I푧푧풙1,푖 − 푚2풓1,푖 × 품 0 −푎3 푎2 + 푚2풓1,푖 × 퐯̇푏 + 푚2풓1,푖 × 훚̇ 푏 × 풓푖 (46) 퐚̃ = [ 푎3 0 −푎1] (62) + 푚2풓1,푖 × 훚푏 × 훚푏 × 풓푖 −푎2 푎1 0 + 푚2풓1,푖 × 훚̇ 푏 × 풓1,푖 + 푚2풓1,푖 × 훚푏 × 훚푏 × 풓1,푖 Therefore, the torque equation over the base body is then expressed as, The equation of forces is obtained after replacing (45) in (35). This preliminary result is simplified if the relationship 푏 푏 푏 푏 푏 푏 푏 훕푒푥푡 = 퐈푏 훚̇ 푏 + 훚̃ 푏 퐈푏 훚푏 + 퐼푦푦 퐘1휃̈ for 풓푖 is used in conjunction with the fact that for a 푏 푏 푏 + 휃̇2퐼푧푧 퐗1휃̇ + 퐼푦푦 훚̃ 푏 퐘1휃̇ symmetrical 4-CMG the vectors 풓1,푖 are, 4 (63) 푏 푏 + 휃̇2퐼푧푧 훚̃ 푏 ∑ 퐳1,푖 퐫 = −퐫 1,0 1,2 (47) 푖=1 퐫1,1 = −퐫1,3 (48) 푏 푏 푏 푏 Where 퐗1, 퐘1, 퐳1,푖 and 퐑푖 are, The final equation is, 퐅푒푥푡 = (푚0 + 4푚2)퐯̇푏 − (푚0 + 4푚2)품 (49) 푠휃1 푐훽푐휃2 −푠휃3 −푐훽푐휃4 푏 퐗1 = [−푐훽푐휃1 푠휃2 푐훽푐휃3 −푠휃4 ] (64) Before computing the dynamic equation for moments, the 푠훽푐휃1 푠훽푐휃2 푠훽푐휃3 푠훽푐휃4 expression 푛10,푖 + 퐫풊 × 퐟10,푖 is simplified by using the following relations, 0 −푠훽 0 푠훽 푏 퐘1 = [푠훽 0 −푠훽 0] (65) ′ 퐫푖 = 풓푖 + 풓1,푖 (50) 푐훽 푐훽 푐훽 푐훽 ퟒ ′ 푏 푇 ∑ 푚2퐫푖 × 품 = 0 (51) 퐳1,1 = [−푐휃1 푐훽푠휃1 푠훽푠휃1] (66) 풊=ퟎ 푏 푇 퐳1,2 = [푐훽푠휃2 푐휃2 푠훽푠휃2] (67)

45 Yime-Rodríguez et al / DYNA 81 (184), pp. 41-47. June, 2014.

푏 푇 퐳1,3 = [푐휃3 푐훽푠휃3 푠훽푠휃3] (68) Let us assume a flywheel with inertia of 0.16 in x and y axis 푏 푇 퐳1,4 = [푐훽푠휃4 −푐휃4 푠훽푠휃4] (69) and a value of 0.308 in z axis, which is rotating at a speed of 10000 rpm. If the body has an angular velocity and 푠휃1 0 −푐휃1 acceleration, it is clear in eq. (74) than the first two terms 푏 퐑1 = [−푐훽푐휃1 푠훽 −푐훽푠휃1] (70) contribute to the total torque. Fig. 5 illustrates the results 푠훽푐휃1 푐훽 푠훽푠휃1 obtained for a unit angular velocity and acceleration. The continuous line represents the torque computed with eq. (74), 푐훽푠휃2 −푠훽 푐훽푠휃2 while the dotted line is the torque computed using the 푏 퐑2 = [ 푠휃2 0 푐휃2 ] (71) traditional approach. 푠훽푐휃2 푐훽 푠훽푠휃2 4. CONCLUSIONS −푠휃 0 푐휃 3 3 푏퐑 = [푐훽푐휃 −푠훽 푐훽푠휃 ] (71) 3 3 3 A new dynamic model for a 4-CMG was derived using 푠훽푐휃3 푐훽 푠훽푠휃3 the Newton-Euler algorithm, a methodology commonly used in Robotics. Although some simplifications were done, the −푐훽푠휃4 푠훽 −푐훽푠휃4 푏 dynamic model is useful to study the behavior of a 4-CMG. 퐑4 = [ −푠휃4 0 −푐휃4 ] (71) 푠훽푐휃4 푐훽 푠훽푠휃4 The obtained dynamic model can also be used for computing a control law for a 4-CMG. Torque equations for the Finally, the torque eq. (34) and (29), can be rearranged rotational joints were also found. A simulation was and expressed in base body coordinates as, performed to illustrate the benefit of the proposed equations. These equations are also useful to compute and help in 푏 푇 푏 푖 푏 푇 풃 훕2,푖 = 퐳1,푖 퐑푖 퐈푖 퐑푖 훚̇ 푏 selecting the proper motors that will drive the joints. 푏 푇 푏 푏 푖 푏 푇 풃 + 퐳1,푖 훚̃ 푏 퐑푖 퐈푖 퐑푖 훚푏 (74) References + 휃̇ I 푏퐳푇 푏훚̃ 푏풚 푖 푦푦 1,푖 푏 1,푖 [1] Kuhns, M. and Rodriguez, A. Singularity avoidance control laws for a 푏 푇 푏 푖 푏 푇 풃 훕1,푖 = 퐲1,푖 퐑푖 퐈푖 퐑푖 훚̇ 푏 multiple CMG system, Proceedings of the 푏 푇 푏 푏 푖 푏 푇 풃 American Control Conference (ACC), pp. 2892–2893, 1994. + 퐲1,푖 훚̃ 푏 퐑푖 퐈푖 퐑푖 훚푏 (75) ̇ 푏 푇 푏 푏 ̈ + 휃2,푖I푧푧 퐲1,푖 훚̃ 푏 풛1,푖 + 휃푖I푦푦 [2] Kuhns, M. and Rodriguez, A. A preferred trajectory tracking steering law for spacecraft with redundant CMGS, Proceedings of the American Control Conference (ACC), pp. 3111–3115, 1995. These equations are useful to select the motors for each actuated joint [13]. [3] Oh, S. and Vadali, S. R. Feedback control and steering laws for spacecraft using single gimbal control moment gyros, Astronautical Sciences, 39 (2), pp. 183–203, 1991. 3. Numerical example [4] Thornton, B., Ura, T., Nose, Y. and Turnock, S. Internal actuation of underwater robots using control moment gyros, Proceedings of Oceans, pp The eq. (74) and (75) are useful for computing the motor 591–598, 2005. requirements, while equation (63) is used to create a steering control law for the 4-CMG as is done in [14]. In the case of a [5] Thornton, B., Ura, T. and Nose, Y. Wind-up AUVs: Combined energy storage and attitude control using control moment gyros. Proceedings of flywheel motor, eq. (74), only the last term is traditionally Oceans, pp. 1-9, 2007 taken into account to compute the required torque, but a numerical simulation can show how the proposed equations [6] Lappas, V. J., Steyn, W. H. and Underwood, C. I. Torque amplification of control moment gyros, IEEE Electronics Letters, 38 (15), pp. 837–839, are better than the traditional approach. 2002.

[7] Tekinalp, O., Elmas, T. and Yavrucuk, I. Gimbal angle restricted control moment gyroscope clusters, Proceedings of 4th International Conference on Recent Advances in Space Technologies (RAST), pp. 585-590, 2009. [8] Ford, K. A. and Hall, C. D. Singular direction avoidance steering for control-moment gyros, Journal Guidance Control and Dynamics, 23 (4), pp. 648–656, 2000. [9] Bedrossian, N. S., Paradiso, J., Bergmann, E. V. and Rowell, D. Redundant single gimbal control moment gyroscope singularity analysis, Journal Guidance Control and Dynamics, 13 (6), pp. 1096–1101, 1990. [10] Toz, M. and Kucuk, S. A comparative study for computational cost of fundamental robot manipulators, Proceedings of International Conference on Industrial Technology (ICIT), pp. 289-293, 2011. [11] Negrean, I., Schonstein, C., Negrean, D.C., Negrean, A.S. and Duca, A.V. Formulations in robotics based on variational principles, Proceeding of International Conference on Automation Quality and Testing Robotics

Figure 5. Effect of body motion in flywheel torque. (AQTR), pp. 1-6, 2010.

46 Yime-Rodríguez et al / DYNA 81 (184), pp. 41-47. June, 2014.

[12] Tsai, L. W. Robot Analysis: The Mechanics of Serial and Parallel C. A. Peña-Cortés has been a Professor at Universidad de Pamplona in the Manipulators. Jon Wiley & Sons, Inc., 1999. Department of Mechatronics, Mechanics and Industrial Engineering since 2004. He received the Electromechanic Engineering degree in 2001 from [13] Jaramillo, A. Franco, E. Guasch, L. Estimación de parámetros Universidad Pedagógica y Tecnológica de Colombia, an MSc Degree in invariantes para un motor de inducción. Dyna, 78 (169), pp. 88–94, 2011. Electronics and Computers Engineering in 2003 from Universidad de los [14] Yime, E., Quintero, J., Saltaren R. and Aracil R. A new Approach for Andes, and his PhD degree in Automatics and Robotics from Universidad Avoiding Internal Singularities in CMG with Pyramidal Shape Using Sliding Politécnica de Madrid in 2009. His researches are focused on Service Control. Proceedings of European Control Conference (ECC), pp. 125-132, Robots, Telerobotics and Parallel Robots. 2009. W. M. Rojas-Contreras is full professor of the Pamplona University. He E. Yime-Rodríguez is a Mechanical Engineer, graduated from Universidad became a systems engineer in Francisco de Paula Santander University. He del Norte, with a PhD in robotics, from Universidad Politecnica de Madrid, received specialist degree from Industrial University of Santander. He who actually work for Universidad Tecnologica de Bolivar, located in obtained a master degree in computer science from Autonomous University. Cartagena, Colombia. His major research areas are parallel and serial He is candidate to PhD in Applied Sciences from Andes University. Now, robotics, with emphasis in kinematics, dynamics and nonlinear control. he is Dean of Engineering and Architecture Faculty and head of Computer Recently he has gained significant experience in mechanical and Science research Group. His researches are focused on software engineering mechatronics design applied to robotics. He can be contacted to and project management software. [email protected]

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