The 14th IFToMM Word Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.PS3.010 Spring Gravity Compensation Using the Noncircuar Puey and Cabe For the Less-Spring Design M.C. Cui 1 S.X. Wang 2 J.M. Li 3 Tianjin University Tianjin University Tianjin University Tianjin, China Tianjin, China Tianjin, China Abstract: This work proposed a new method for a gravity compensation using the noncircuar puey, cabe driving, and spring baancer. The noncircuar puey is arranged in the front of one transmission chain. When the mechanism appy this method for the gravity compensation of some muti-dof and mutiink manipuator, it can be achieved that a perfect gravity baance and ess number of springs. To obtain the parameter and arrangement of the spring and transmission, the potentia energy reated to the transmission structure is discussed. Based on the invariance of a the potentia energy of a static baance mechanism, some design rues and cases are presented to get the transmission function. In this study, the synthesis of the noncircuar puey is demonstrated according to the estabished transmission function. At ast, the exampe of gravity compensation for two-dof yaw-pitch mechanism is presented. Keywords: Gravity compensation, Spring baancer, Cabe driven, Noncircuar puey I. Introduction Gravity compensation of a mechanism can faciitate contro and improve energy efficiency. There are many avaiabe ways to satisfy the need of gravity compensation. For the static baancing, the method of spring baancer is advantageous in the introduction of oy sma amount of additiona inertia to the origina system. Many systems, such as passive rehabiitation devices, service robot, fight simuator, industria robot, use the spring baancer to improve their motion characteristics. For severa decades, many kinds of gravity compensation with springs have been proposed. Urich and Kumar [1] presented a one-dof (degree of freedom) gravity compensator comprised of a noncircuar puey, wire and spring. Simiary, Endo [2] presented a proximate spring baancing used a noncircuar puey and a spring for a three-dof mechanism. I. Simionescu [3, 4] proposed the discrete and continuous baancing using the muti-ink and noncircuar puey transmission for one-dof spring gravity compensation. Kim [5] presented the perfect muti-dof gravity compensation with severa one-dof spring baancers. In this design, the one-dof spring baancer was aso comprised of the noncircuar puey and spring. Rahman [6] presented a one-dof spring baancer using zero-free-ength spring for a singe ink and extended it to baance a muti-ink with the auxiiary inks. Agrawa and Fattah [7, 8] proposed a method for a muti-ink panar or 1 cuimengchao@tju.edu.cn 2 shuxinw@ tju.edu.cn 3 jimmyzhq@gmai.com spatia manipuator. In their study, paraeogram was adopted to represent the COM (center of mass), and springs were equipped at the paraeogram. Deepak and Ananthasuresh [9], Lin [10, 11] presented perfect spring compensation using oy springs without the hep of auxiiary inks and respectivey deveoped synthesis methods for muti-ink gravity compensation. Cho [12] presented muti-dof gravity compensation using beve gears and one-dof spring baancers. Herder [13] presented the spring to spring baancer which can adapt varying oad. Gossein [14] presented spring baancers for parae mechanisms. Athough gravity compensation has variety of types, a unit of spring and transmission are existed in most designs. This work was inspired by the previous work [1~4, 10] using the simiar structures comprised of the transmission and one-dof spring baancer. The transmission has severa different types such as auxiiary inks, cabe (bet) driving, beve gears. However, the specia structure, the paraeogram and the puey of same radius, causes the expression of potentia energy of each spring baancer have a common characteristic. It iustrates the mathematic reationship of rotation ange of each joint is inear sum in one transmission chain. Our work is to change this inear transmission into a more compicated situation by using the noncircuar puey instead of circuar puey. Different from the reach [1~5], the noncircuar puey is arranged in the start of one transmission chain. A compicated transmission structure which aso has more freedom to satisfy the counterforce provides more avaiabiity to the new design of the perfect gravity compensation. To obtain the coefficient and arrangement of the spring and the transmission, the tota potentia energy of the springs and the manipuator mass has been investigated. The perfect baance is equivaent to invariance of a the potentia energy of springs and mass with respect to a configurations of the mechanism. Agrawa and Fattah [7, 8] presented a hybrid strategy with the hep of auxiiary inks based on the expression of potentia energy. Lin [10, 11] proposed the stiffness bock method to determine the spring s arrangement with no auxiiary inks. Cho [12] proposed the mapping matrix to the design of the transmission between the spring baancer and inkage. Our method is to anayze the expression of the potentia energy using the noncircuar puey transmission structure. Based on the principe of the invariance of the potentia energy, the gravitationa potentia energy of the manipuator is divided into the severa parts which are corresponded to the new kind of spring baancer. Some design rues and cases are demonstrated using the noncircuar puey to achieve a perfect and ess spring number gravity compensation.
We impement the theory about the design of the noncircuar puey on foundations of the former study [5]. According to the function of the transmission, the circe and noncircuar puey are arranged in a seria by cabe driven. Then, the geometry curve of the noncircuar puey is cacuated using the infinitesima mathematic process. The needed structure of antagonistic cabe driven and the assembe puey are demonstrated. This paper is organized as foows: Section 2 describes the genera expression of potentia energy in spring baance mechanism using the paraeogram and circe puey. Section 3 presents the transmission structure and the expression of potentia energy using the noncircuar puey. Then some design rues and cases are demonstrated for the spring baance mechanism. Section 4 presents the synthesis of the noncircuar puey. Section 5 presents exampe to verify the effectiveness and advantage of this method which makes the spring baance mechanism have ess number of springs. Section 6 presents concusion of the present work. II. Spring compensation with inear transmission The most spring gravity compensation for muti-dof and muti-ink manipuator consists of the spring baancer and the transmission. Each spring provides a counterbaance with the characteristic of the transmission. As a resut, the entire potentia energy of the ink mass is divided into severa specified parts baanced by the spring. This section introduces the basic theory of the spring compensation using the paraeogram or circe puey. Some gravity compensations are discussed to show the feature of the transmission. These exampes are defined as the spring compensations with inear transmission. U s = C mg cos θ (3) Here U m denotes the potentia energy of the ink mass, U s denotes the potentia energy of the spring, m is the ink mass, g is the acceeration of gravity, is the ength of the weight arm, θ is the ange between the vertica and the weight arm. In spite of different structures of the spring baancer, the eastic potentia energy of each spring has the same function form of the joint ange for one certain mechanism. B. Muti-ink and muti-dof spring gravity compensation For the muti-ink or muti-dof manipuator, the gravitationa potentia energy is the sum of severa different energy items which are the trigonometric terms. The singe spring using a transmission structure generates the coupe torques on each joint. The chaenge of a perfect gravity compensation is to generate proper counterbaance torque using the most est springs and transmissions. The muti-dof or muti-ink staticay baancing system aso obeys the principe, the invariance of the entire potentia energy. With a transmission structure, the eastic potentia energy expression of one spring has a specific form. Because of the compex situation that the expression of the gravitationa potentia energy is the function with respect to muti joint ange, the singe spring cannot finish the work. Severa different springs make up the gravity compensation for the muti-dof and muti-ink manipuator. A. One-DOF spring gravity compensation Fig. 2. Two-DOF two-ink gravity compensation: (a) using the zero-free-ength spring and cabe driven (b) using the noncircuar puey and paraeogram Considering a two-ink two-dof mechanism, there are some methods to baance the gravity of ink mass shown in Fig. 2. The gravitationa potentia energy of tota masses is computed by Fig. 1. One-DOF spring baancer: (a) using the zero-free-ength spring (b) using the noncircuar puey One-DOF spring gravity compensator is used for gravity baancing of the mechanism which has one horizonta rotation axis. As shown in Fig. 1, the one-dof spring gravity compensator is mosty made of the two different base structures, the zero-free-ength spring or the noinear circuar puey. When the mechanism system is gravity baancing, the tota potentia energy is constant. This impies that, the expressions of gravitationa potentia energy and eastic potentia energy with respect to the joint ange are compementary. For the one-dof mechanism baanced by one spring in Fig. 1, this can be written as U m + U s = Constant (1) U m = mg cos θ (2) U m = (m 1 g 1 + m 2 gl 1 ) cos θ 1 +m 2 g 2 cos(θ 1 + θ 2 ) Here m i is the mass of ink i, i is the ength of the weight arm of ink i, θ i is the rotationa ange of the joint i. L 1 is the ength of ink 1. To baance the gravity, the system needs the counterbaancing springs with the potentia energy that varies in the form, cos θ 1 and cos(θ 1 + θ 2 ). Therefore the cos θ 1 denotes a spring baancer with the transmission reated to joint 1. And the cos(θ 1 + θ 2 ) denotes a spring baancer with the transmission reated to joint 1 and joint 2. The paraeogram and cabe driven as the transmission can easiy satisfy the function, θ 1 + θ 2. Athough there are many types for one certain mechanism due to the variety of one-dof spring baancer and the transmission, the each spring finay has the same expression, just as (4)
U s1 = C mg cos θ 1 (5) U s2 = C mg cos(θ 1 + θ 2 ) (6) noncircuar puey transmission has more freedom than muti-ink transmission to generate the functiona potentia energy and achieve the perfect baancing. Noinear transmission can be presented by repacing the circuar puey with the noncircuar puey. As shown in Fig. 4, the mathematics expression of noinear transmission is discussed beow. Fig. 3. (a) One-ink ro-pitch manipuator (b) Gravity compensation using the beve gear in [12] A one-ink ro pitch manipuator is depicted in Fig. 3. Respectivey, the gravitationa potentia energy of the masses is computed by U m = m 1 g 1 cos θ 1 cos θ 2 (7) The cos θ 1 cos θ 2 cannot directy match the structure made up of the spring baancer and transmission. It was aways transformed through the product-to-sum identity cos θ 1 cos θ 2 = 0.5 cos(θ 1 + θ 2 ) + 0.5 cos(θ 1 θ 2 ) (8) This equation represents the transmission is simiar to a mechanica differentia to satisfy the both θ 1 + θ 2 and θ 1 θ 2. The mechanica differentia, the gear or cabe driven differentia, connects with the two spring baancers to compensate the gravity. These springs generate correspondent eastic potentia energy U s1 = C 0.5m 1 g 1 cos(θ 1 + θ 2 ) (9) U s2 = C 0.5m 1 g 1 cos(θ 1 θ 2 ) (10) As a summary, these cacuations show that muti-ink and muti-dof spring gravity compensations are the system of severa spring baancers and transmission. The mathematic function of these transmissions is the form of the inear sum of each joint ange. This impies that the gravitationa potentia energy shoud be divided into different energy bases according to the transmission structure. That is to say, the transmission decides the method of muti-ink and muti-dof spring gravity compensation. To obtain a new method, one way is to change the transmission foowed by the change of its expression. III. Spring compensation with noinear transmission This section introduces the transmission using the noncircuar puey which is defined as noinear transmission. Spring baancer with a noinear transmission provides an eastic potentia energy of new expression. The method of fabricating the possibe noinear transmission for gravity compensation is proposed. Fig. 4. Cabe driven using the noncircuar puey The Fig. 4 shows a seria of cabe transmission structure. With respect to the every figure, the reationship of the rotation ange between the first puey and inks can be described as foows θ 0 p = f 1 (θ 1 ) (11) θ 1 p = θ 1 + f 2 (θ 2 ) (12) θ 1 p = θ 1 + f 2 (θ 2 + f 3 (θ 3 + )) (13) Here θ p i denotes the rotation ange of the puey i, θ i denotes the rotation ange of the ink i, the mapping f i presents the rotation ange of the puey i 1 caused by the rotation ange of the puey i. In Fig. 4(a), the rotation of puey 0 is oy caused by the rotation of the puey 1 which is the same as the rotation of the ink 1. In Fig. 4(b), the rotation of puey 1 is caused by the rotation of puey 2 and the rotation of ink 1. It can be expressed as θ p 1 = θ 1 + f 2 (θ p 2 ) θ p 2 = θ 2. It can be expanded into the genera p expression as θ i 1 = θ i + f i (θ p i 1 ). By combining a the equation of the rotation anges of each adjacent puey, Eq. (13) can be concuded as the genera expression for noinear transmission. B. Potentia energy with noinear transmission The energy base which presents the gravitationa potentia energy baanced by one spring is determined by transmission structure. To distinguish the two kinds of the energy base, the energy base decided by the inear transmission is defined as the inear energy base. Reativey, the energy base decided by the noinear transmission is defined as the noinear energy base. A. Noinear transmission To adjust the mapping between the input and output, specia transmission is used in mechanism. I. Simionescu [3, 4] deveoped the discrete and continuous transmission using the muti-ink and noncircuar puey for gravity compensation. The aim of these designs is to modify the motion of springs to match the motion of the ink. The
base. Eq. (26) and Eq. (27) present two equations with two undetermined mapping f 21 and f 22. This impies F 1 (θ 2 ) and F 2 (θ 2 ) has more freedom to match the expression of the entire potentia energy. Then the sum of two inear energy bases, U 1 = cos (θ 1 + θ 2 ) and U 2 = cos (θ 1 θ 2 ) can be expressed as Fig. 5. One-DOF spring baancer connected with cabe driven transmission Consider that two kinds of the transmission connect with the one-dof spring baancer as shown in Fig. 5. The genera expression of potentia energy base can be expressed as U j = cos (θ 1 ± ±θ n ) (14) U j = cos (θ 1 ± f 2j (θ 2 ± f 3j (θ 3 ± ))) (15) Here U j denotes the inear energy base baanced by the spring j, U j denotes the noinear energy base baanced by the spring j, f nj denotes the rotation mapping in transmission connected with spring j. Notice that the mapping f nj is deveoped by the requirement, U j presents a more variabe function than U j. When the mapping f nj is equa to 1, U j and U j are same. C. Methodoogy for the transmission function Athough there are two kinds of energy bases, the sum of inear or noinear energy bases must be equa to the entire gravitationa potentia energy for one staticay baancing mechanism. This impies that the sum of some inear energy bases must be equa to the sum of some noinear energy bases. This reationship can be described as V m = U j = U j (16) The question to obtain the category of noinear transmission is equivaent to the combination category of the noinear energy bases. Considering a noinear energy base with two pueys in a genera situation, it can be expressed as U i = cos(θ 1 + f i (θ 2 )) (17) U i = cos θ 1 cos(f i (θ 2 )) sin θ 1 sin(f i (θ 2 )) (18) We assume that the mapping f i is not equa to 1. Notice that the noinear energy base cannot be independenty equa to any one inear energy base, just as cos (θ 1 + θ 2 ), cos (θ 1 θ 2 ), cos (θ 1 ). The sum of any two noinear energy bases with two pueys can be expressed as K 1 U 1 + K 2 U 2 = cos θ 1 F 1 (θ 2 ) sin θ 1 F 2 (θ 2 ) (19) F 1 (θ 2 ) = K 1 cos(f 1 (θ 2 )) + K 2 cos(f 2 (θ 2 )) (20) F 2 (θ 2 ) = K 1 sin(f 1 (θ 2 )) + K 2 sin(f 2 (θ 2 )) (21) Here K i denotes the parameter of one noinear energy K 1 U 1 + K 2 U 2 = cos θ 1 F 1 (θ 2 ) sin θ 1 F 2 (θ 2 ) (22) F 1 (θ 2 ) = K 1 cos(θ 2 ) + K 2 cos(θ 2 ) (23) F 2 (θ 2 ) = K 1 sin(θ 2 ) K 2 sin(θ 2 ) (24) Here K i denotes the parameter of one inear energy base and K i is known decided by the mechanism. According to the reationship of the eastic potentia energy for one certain mechanism, Eq. (16) requires that the K 1 U 1 + K 2 U 2 is equa to the K 1 U 1 + K 2 U 2. Therefore the reationship between F i (θ 2 ) and F i (θ 2 ) is expressed as F 1 (θ 2 ) = F 1 (θ 2 ) (25) F 2 (θ 2 ) = F 2 (θ 2 ) (26) The Eq. (25) and Eq. (26), two noinear equations with respect to the θ 2, determine the two unknown mappings and two unknown parameter K i. It is obvious that the four unknown quantity can be cacuated. The foowing contents describe some cases to achieve the transformation. Fig. 6. Muti-DOF or Muti-ink manipuator Case I: The noinear energy base, such as cos(f 1 (θ 1 )), is a function about singe joint ange θ 1. For some 1-DOF muti-ink mechanism, the entire gravitationa potentia energy cannot be simpified to the form as cos θ. Cho [15] presents a design method for static baancer with the associated inkage. In his reach, various gravity compensations are designed for four-bar inkage and sider crank. Severa springs and paraeogram are adopted, but spring numbers are more than their DOFs. For 1-Dof cose-oop mechanism to reduce spring number, we assume a the gravitationa torques caused by inkage mass effect on the singe joint. Because the other joint motion is correated with this singe joint, the potentia energy can be expanded as one variabe function.
As shown in Fig. 6(a), it is described as V m = A cos θ 1 + B cos(θ 1 + θ 2 ) +C cos(θ 1 + θ 2 + θ 3 ) A = m 1 g 1 + (m 2 + m 3 )gl 1 (27) B = m 2 g 2 + m 3 gl 2 C = m 3 g 3 (28) Notice that the ange θ 2 and θ 3 is reated to θ 1. The Eq. (27) can be written as V m = A cos θ 1 + B cos(f a (θ 1 )) + C cos(f b (θ 1 )) (29) f a (θ 1 ) = θ 1 + θ 2 f b (θ 1 ) = θ 1 + θ 2 + θ 3 (30) Here f a and f b denote the mapping between these anges. Furthermore the Eq. (29) is described as the desired form which represents the noinear energy base as V m = K cos(f s (θ 1 )) (31) The Eq. (31) shows that one-dof four-bar mechanism can be baanced by one noinear transmission and one spring. Case II: When the mechanism in Fig. 6(a) exists as the part of entire mechanism, the gravity compensation can be aso designed using the noinear transmission for muti-dof mechanism. As shown in Fig. 6(b), the potentia energy can be described as V m = A cos θ 1 + B cos(θ 1 + θ 2 ) +C cos(θ 1 + θ 2 + θ 3 ) +D cos(θ 1 + θ 2 + θ 3 + θ 4 ) A = m 1 g 1 + (m 2 + m 3 + m 4 )gl 1 B = m 2 g 2 + m 3 gl 2 + m 4 gl 2 C = m 3 g 3 + m 4 gl 3 D = m 4 g 4 (32) (33) Notice that the ange θ 3 and θ 4 is oy reated to θ 2. The Eq. (32) can be aso written as V m = A cos θ 1 + B cos(θ 1 + θ 2 ) +C cos(θ 1 + f a (θ 2 )) + D cos(θ 1 + f b (θ 2 )) (34) f a (θ 2 ) = θ 2 + θ 3 f b (θ 2 ) = θ 2 + θ 3 + θ 4 (35) The Eq. (31) shows that the potentia energy of the entire mechanism is the function with respect to θ 1 and θ 2. According to the method from Eq. (16) to Eq. (23), the Eq. (34) can be written as V m = K 1 cos(θ 1 + f 1 (θ 2 )) +K 2 cos(θ 1 + f 2 (θ 2 )) (36) The Eq. (36) shows that mechanism depicted in Fig. 6(b) can be baanced by two noinear transmissions and two springs. Case III: The mechanism shown in Fig. 6(c) is simiar to the mechanism shown in Fig. 6(b). And the potentia energy can be described as V m = A cos θ 1 cos θ 2 + B cos θ 1 cos (θ 2 + θ 3 ) +C cos θ 1 cos (θ 2 + θ 3 + θ 4 ) A = m 1 g 1 + (m 2 + m 3 )gl 1 (37) B = m 2 g 2 + m 3 gl 2 C = m 3 g 3 (38) Notice that the ange θ 3 and θ 4 is oy reated to θ 2. Using the same transforming above, the Eq. (37) can be aso written as V m = A cos θ 1 cos θ 2 + B cos θ 1 cos (f a (θ 2 )) +C cos θ 1 cos(f b (θ 2 )) (39) f a (θ 2 ) = θ 2 + θ 3 f b (θ 2 ) = θ 2 + θ 3 + θ 4 (40) Furthermore the Eq. (39) is expressed as V m = K cos(θ 1 + f(θ 2 )) +K cos(θ 1 f(θ 2 )) 2K cos(f(θ 2 )) = A cos θ 2 + B cos(f a (θ 2 )) +C cos(f b (θ 2 )) (41) (42) Case IV: As shown in Fig. 6(d), the expression of gravitationa potentia energy of the two-dof pitch-yaw mechanism can be expanded as V m = A cos θ 1 + B cos θ 1 cos θ 2 (43) A = m 1 g 1 + m 2 gl B = m 2 g 2 (44) Furthermore the Eq. (43) is extended as V m = K cos(θ 1 + f(θ 2 )) +K cos(θ 1 f(θ 2 )) (45) 2K cos(f(θ 2 )) = A + B cos(θ 2 ) (46) As a summary, the gravity compensation using the
noinear transmission needs ess spring for some specia mechanisms. However this method cannot be used in a the mechanism to reduce spring number for perfect gravity compensation. It depends on the transformation of the expression of gravitationa potentia energy. Some items, just as constant vaue or one variabe function, can be combined together into one item. The design to reaize this mathematic procedure is to change the transmission mapping and the curve of the puey. r a (θ a ) r b (θ a ) = f (θ a ) (52) IV. Synthesis of the noncircuar puey A. Cacuation for the curve of noncircuar puey The transmission mapping f is determined by the procedure mentioned in above chapter. The geometry curve of the noncircuar puey depends on the certain mapping f. Due to noinearity of the transmission mapping, the synthesis of noncircuar puey is based on the infinitesima cacuus approach which eads to an anaytica soution. With the hep of former study [3], there needs a series of cacuations step by step: 1. the ength of the moment arm, 2. the position of the cabe, 3. the geometry curve of the noncircuar puey. In this paper the moment arm is obtained through the transmission mapping f. Fig. 7. A schematic diagram of the transmission with the noncircuar puey and cabe The first step is to derive the reationship between the moment arms of two adjacent pueys by using the principe of virtua work. As shown in Fig. 7(a), the puey a with driving torque τ a drives the puey b with the oad torque τ b in a constant speed by cabe driven. So the virtua work of the system and the reationship between the motions of two pueys can be expressed as τ a δθ a = τ b δθ b (47) f(θ a ) = θ b (48) f (θ a )θ a = θ b (49) Because the torque is equa to the product of the moment arm and the cabe tension, it is described as τ a = F cabe r a (θ a ) (50) τ b = F cabe r b (θ a ) (51) Here, F cabe denotes the tension in cabe, r i (θ a ) denotes the moment arm of the puey i with ange variabe θ a. Considering δθ is equa to θ, the overa procedure derives the reationship between the transmission mapping and the radius of the puey. Fig. 8. A schematic diagram of the geometry curve of the noncircuar puey The second step is to obtain the panar position of the cabe on one puey with respect to a configurations. The radius of puey a is determined by f (θ a ) and r b (θ a ) at each ange of the puey a. For simpification the puey b is considered as circe puey, so that the radius and the moment arm are constant. As shown in Fig. 7(a), the equation of the ine AB which presents the cabe s position at the joint ange of θ a can be expressed as y = A(θ)x + B(θ) (53) A(θ) = tan(θ + μ) (54) B(θ) = r a (cos(θ + μ) sin(θ + μ) tan(θ + μ)) (55) μ = arcsin ((r a (θ) r b (θ)) o ) (56) Here o denotes the distance between the two pueys. The third step is to obtain the equations of the geometry curve. Once the moment arms of each puey are known at the entire possibe anges, the panar positions of the transmission cabe trance out the profie of the puey. As shown in Fig. 7(b), the position of the transmission cabe changes with respect to the motion of puey a. When the θ changes infinitesimay, the intersection point P of the two ines at joint ange θ and θ is at the profie of the puey. So the position of the point P can expressed as x p = (B(θ ) B(θ)) (A(θ ) A(θ)) (57) y p = (A(θ )B(θ) A(θ)B(θ )) (A(θ ) A(θ)) (58) θ = θ + δθ, x p and y p are the coordinates of the intersection point P. When the δθ becomes infinitesima, Eq. (57) and Eq. (58) can be expanded as x p (θ) = B (θ) A (θ) (59)
y p (θ) = A(θ)B (θ) A (θ) + B(θ) (60) B. Parameter seection It is to be necessary that the profie of the puey shoud be a continuous differentiabe curve to reach the proper transmission. In the research [16], it decided that the transmission mapping cannot be a the required functions. Once the derivation of the transmission mapping, f (θ), changes sign, the curve is a strong noinear that eads the interference of the cabe. To avoid this situation, the transmission mapping shoud be seected to generate the proper profie of the puey. Fig. 10. The antagonistic design for cabe driven Different from the circe puey, two ines of the noncircuar puey cannot connect with each other by end to end in most cases. As shown in Fig. 10, the profie of the noncircuar puey consists of two ines separatey which overaps each other. Notice that the geometry of two ines is the same ocated at different anges. It eads to that the noncircuar puey can be assembed with two same pueys in two ayers. Fig. 9. Geometry curve of the noncircuar puey with different parameter K m 1 m 2 1 2 L 1 0.3kg 0.5kg 0.2m 0.2m 0.3m r b o g θ 2 10cm 30cm 0.98 [ 135 o, 135 o ] V. Design exampes This section presents a design exampe of spring gravity compensation for two-dof pitch-yaw manipuator. The new design method using the noncircuar puey wi reduce the number of the springs and abbreviate the entire structure. The transmission structure and the profie of the each puey are discussed and cacuated in more detais. Tabe. 1. The design parameters For exampe, the transmission function of Case IV in Section III is K cos(f(θ 2 )) = m 1 g 1 + m 2 gl + m 2 g 2 cos θ 2. The vaue of the parameter K chosen in a range decides both the geometry curve of the noncircuar puey and motion range of the joint. As shown in Fig. 9, three geometry curves are demonstrated with the parameters in Tabe. 1. Consider that T = m 1 g 1 + m 2 gl + m 2 g 2. When the vaue of K is equa to T, the curve is a continuous ine and the motion range is wide enough from π to π. When the K is ess or great than T, the curve is discrete and the motion range is decreasing. C. The design of the antagonistic cabe driven In many robots, the gravitationa torque is bi-directiona effect on the joint. To obtain more motion space with the configuration of the gravity compensation, the antagonistic cabe driven design is necessary to transmit bi-directiona torque. According to cose-oop transmission of the cabe driven using the circe puey, we consider that there are two ines on one puey which shoud be cacuated separatey. When the cabe wraps a distance around the puey on the one side, the cabe unwraps the same distance on the other side. The antagonistic design of the noncircuar puey is to maintain the ength of each moment arm equa on each side. Fig. 11. Two-DOF pitch-yaw manipuator The manipuator is depicted in Fig. 11. θ 1 and θ 2 represent rotation anges in the z 1 and z 2 axes, respectivey. The gravitationa potentia energy is computed by V m = V m1 + V m2 V m1 = m 1 g 1 cos θ 1 V m2 = m 2 g 2 cos θ 1 cos θ 2 + m 2 gl 1 cos θ 1 i denotes the moment arm of the ink i, L i denotes the distance of the ink. Notice that the entire potentia energy has two energy items of cos θ 1 cos θ 2 and cos θ 1 which can generate three inear energy bases, U 1 = K 1 cos θ 1 U 2 = K 2 cos(θ 1 + θ 2 ) U 3 = K 3 cos(θ 1 θ 2 )
K i denotes the parameter of the inear energy base i. So K 1 = m 1 g 1 + m 2 gl 1, K 2 = K 3 = 0.5m 2 g 2. Three inear energy bases denote the gravity compensation of this manipuator shoud have three transmission chains and three springs using the circe puey or paraeogram. According to the method described in Section III, two noinear energy bases, instead of three inear energy bases, can be fabricated to reduce the number of springs and cabes. For simpicity, the specific expanding is demonstrated as foows 4 15 15 5 15 30 6 30 15 7 30 30 Tabe. 2. Poses for simuation V m = K cos θ 1 cos(f(θ 2 )) K cos(f(θ 2 )) = m 2 g 2 cos θ 2 + m 2 gl 1 + m 1 g 1 Then, the spring compensation system consists of two noinear energy bases, just as U 1 = 0.5K 1 cos(θ 1 + f(θ 2 )) U 2 = 0.5K 2 cos(θ 1 f(θ 2 )) K i denotes the parameter of the noinear energy base i, K 1 = K 2 = K. Fig. 13. Computation of the potentia energy To verify the effectiveness of the method, seven poses are chosen to cacuate the eastic potentia energy and gravitationa potentia energy of the mechanism. The pose index is shown in Tabe. 2. As shown in Fig. 17, the tota potentia energy maintains the invariant for a the poses. (a) (b) Fig. 12. Spring gravity compensation for two-dof pitch-yaw mechanism: (a) panar schematic (b)stereoscopic mode The reevant parameters for the design are chosen as: m 1 = 0.3kg, m 2 = 0.5kg, L 1 = 0.3m, 1 = 2 = 0.2m, r 1 = 0.1m. And K is cacuated to be 0.1514N. m. In Fig. 12, the entire part of the transmission and mechanism is demonstrated by panar schematic and stereoscopic mode. Pose θ 1 θ 1 1 0 0 2 0 15 3 15 0 VI. Concusion of the present work This paper presents a new method for static baancing of mechanisms with conservative oads such as gravity and spring oads using noncircuar puey and cabe driven. The method, which competey baances gravitationa torque, provides reduction the actuator requirements under space constraints. The transmission structure is key factor to determine the expression of the potentia energy of the springs. The new transmission structure, using the noncircuar puey at the start of the transmission, is adopted to generate more fexibe spring force for gravity compensation. The entire gravitationa potentia energy is divided into severa parts which correspond with the new structure. This paper uses a simpe way, transformation of the expression of the potentia energy, to find out the transmission structure and coefficient of the spring. As a resut, the number of the springs is reduced for the gravity compensation of some specia mechanism. Athough the method cannot appy in a the mechanism to reduce the spring number, it is a new way to achieve a compete static baancing. With no auxiiary inkages and suspending cabe, the resutant mechanism is more compact. We impement the theory of the design of the noncircuar puey based on the former study. According to the function of the transmission mapping, the geometry curve of the noncircuar puey is cacuated using the infinitesima mathematic process. The circe puey and noncircuar puey are arranged in a seria by cabe driven. The seection and a needed structure of antagonistic cabe driven are demonstrated. Future work is to find out the more avaiabiity of the transmission structures which present more avaiabiity of
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