The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.PS3.010

Spring Gravity Compensation Using the Noncircular Pulley

and Cable For the Less-Spring Design

M.C. Cui1 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 noncircular pulley, cable

driving, and spring balancer. The noncircular pulley is

arranged in the front of one transmission chain. When the

mechanism apply this method for the gravity compensation

of some multi-DOF and multilink manipulator, it can be

achieved that a perfect gravity balance and less number of

springs. To obtain the parameter and arrangement of the

spring and transmission, the potential energy related to the

transmission structure is discussed. Based on the

invariance of all the potential energy of a static balance

mechanism, some design rules and cases are presented to

get the transmission function. In this study, the synthesis of

the noncircular pulley is demonstrated according to the

established transmission function. At last, the example of

gravity compensation for two-DOF yaw-pitch mechanism

is presented. Keywords: Gravity compensation, Spring balancer, Cable driven,

Noncircular pulley

I. Introduction

Gravity compensation of a mechanism can facilitate

control and improve energy efficiency. There are many

available ways to satisfy the need of gravity compensation.

For the static balancing, the method of spring balancer is

advantageous in the introduction of only small amount of

additional inertia to the original system. Many systems,

such as passive rehabilitation devices, service robot, flight

simulator, industrial robot, use the spring balancer to

improve their motion characteristics.

For several decades, many kinds of gravity

compensation with springs have been proposed. Ulrich and

Kumar [1] presented a one-DOF (degree of freedom)

gravity compensator comprised of a noncircular pulley,

wire and spring. Similarly, Endo [2] presented a proximate

spring balancing used a noncircular pulley and a spring for

a three-DOF mechanism. I. Simionescu [3, 4] proposed the

discrete and continuous balancing using the multi-link and

noncircular pulley transmission for one-DOF spring

gravity compensation. Kim [5] presented the perfect

multi-DOF gravity compensation with several one-DOF

spring balancers. In this design, the one-DOF spring

balancer was also comprised of the noncircular pulley and

spring.

Rahman [6] presented a one-DOF spring balancer using

zero-free-length spring for a single link and extended it to

balance a multi-link with the auxiliary links. Agrawal and

Fattah [7, 8] proposed a method for a multi-link planar or

1cuimengchao@tju.edu.cn 2shuxinw@ tju.edu.cn 3jimmyzhq@gmail.com

spatial manipulator. In their study, parallelogram was

adopted to represent the COM (center of mass), and springs

were equipped at the parallelogram. Deepak and

Ananthasuresh [9], Lin [10, 11] presented perfect spring

compensation using only springs without the help of

auxiliary links and respectively developed synthesis

methods for multi-link gravity compensation. Cho [12]

presented multi-DOF gravity compensation using bevel

gears and one-DOF spring balancers. Herder [13]

presented the spring to spring balancer which can adapt

varying load. Gosselin [14] presented spring balancers for

parallel mechanisms.

Although 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 similar structures comprised of the transmission

and one-DOF spring balancer. The transmission has

several different types such as auxiliary links, cable (belt)

driving, bevel gears. However, the special structure, the

parallelogram and the pulley of same radius, causes the

expression of potential energy of each spring balancer have

a common characteristic. It illustrates the mathematic

relationship of rotation angle of each joint is linear sum in

one transmission chain. Our work is to change this ‘linear’

transmission into a more complicated situation by using the

noncircular pulley instead of circular pulley. Different

from the reach [1~5], the noncircular pulley is arranged in

the start of one transmission chain. A complicated

transmission structure which also has more freedom to

satisfy the counterforce provides more availability to the

new design of the perfect gravity compensation.

To obtain the coefficient and arrangement of the spring

and the transmission, the total potential energy of the

springs and the manipulator mass has been investigated.

The perfect balance is equivalent to invariance of all the

potential energy of springs and mass with respect to all

configurations of the mechanism. Agrawal and Fattah [7, 8]

presented a hybrid strategy with the help of auxiliary links

based on the expression of potential energy. Lin [10, 11]

proposed the stiffness block method to determine the

spring’s arrangement with no auxiliary links. Cho [12]

proposed the mapping matrix to the design of the

transmission between the spring balancer and linkage.

Our method is to analyze the expression of the potential

energy using the noncircular pulley transmission structure.

Based on the principle of the invariance of the potential

energy, the gravitational potential energy of the

manipulator is divided into the several parts which are

corresponded to the new kind of spring balancer. Some

design rules and cases are demonstrated using the

noncircular pulley to achieve a perfect and less spring

number gravity compensation.

We implement the theory about the design of the

noncircular pulley on foundations of the former study [5].

According to the function of the transmission, the circle

and noncircular pulley are arranged in a serial by cable

driven. Then, the geometry curve of the noncircular pulley

is calculated using the infinitesimal mathematic process.

The needed structure of antagonistic cable driven and the

assemble pulley are demonstrated.

This paper is organized as follows: Section 2 describes

the general expression of potential energy in spring balance

mechanism using the parallelogram and circle pulley.

Section 3 presents the transmission structure and the

expression of potential energy using the noncircular pulley.

Then some design rules and cases are demonstrated for the

spring balance mechanism. Section 4 presents the synthesis

of the noncircular pulley. Section 5 presents example to

verify the effectiveness and advantage of this method

which makes the spring balance mechanism have less

number of springs. Section 6 presents conclusion of the

present work.

II. Spring compensation with linear transmission

The most spring gravity compensation for multi-DOF

and multi-link manipulator consists of the spring balancer

and the transmission. Each spring provides a

counterbalance with the characteristic of the transmission.

As a result, the entire potential energy of the link mass is

divided into several specified parts balanced by the spring.

This section introduces the basic theory of the spring

compensation using the parallelogram or circle pulley.

Some gravity compensations are discussed to show the

feature of the transmission. These examples are defined as

the spring compensations with linear transmission.

A. One-DOF spring gravity compensation

Fig. 1. One-DOF spring balancer:

(a) using the zero-free-length spring

(b) using the noncircular pulley

One-DOF spring gravity compensator is used for gravity

balancing of the mechanism which has one horizontal

rotation axis. As shown in Fig. 1, the one-DOF spring

gravity compensator is mostly made of the two different

base structures, the zero-free-length spring or the nonlinear

circular pulley. When the mechanism system is gravity

balancing, the total potential energy is constant. This

implies that, the expressions of gravitational potential

energy and elastic potential energy with respect to the joint

angle are complementary. For the one-DOF mechanism

balanced by one spring in Fig. 1, this can be written as

𝑈𝑚 + 𝑈𝑠 = Constant (1)

𝑈𝑚 = 𝑚𝑔𝑙 cos 𝜃 (2)

𝑈𝑠 = 𝐶 − 𝑚𝑔𝑙 cos 𝜃 (3)

Here 𝑈𝑚 denotes the potential energy of the link mass, 𝑈𝑠

denotes the potential energy of the spring, 𝑚 is the link

mass, 𝑔 is the acceleration of gravity, 𝑙 is the length of the

weight arm, 𝜃 is the angle between the vertical and the

weight arm. In spite of different structures of the spring

balancer, the elastic potential energy of each spring has the

same function form of the joint angle for one certain

mechanism.

B. Multi-link and multi-DOF spring gravity compensation

For the multi-link or multi-DOF manipulator, the

gravitational potential energy is the sum of several

different energy items which are the trigonometric terms.

The single spring using a transmission structure generates

the couple torques on each joint. The challenge of a perfect

gravity compensation is to generate proper counterbalance

torque using the most lest springs and transmissions. The

multi-DOF or multi-link statically balancing system also

obeys the principle, the invariance of the entire potential

energy. With a transmission structure, the elastic potential

energy expression of one spring has a specific form.

Because of the complex situation that the expression of the

gravitational potential energy is the function with respect

to multi joint angle, the single spring cannot finish the

work. Several different springs make up the gravity

compensation for the multi-DOF and multi-link

manipulator.

Fig. 2. Two-DOF two-link gravity compensation:

(a) using the zero-free-length spring and cable driven

(b) using the noncircular pulley and parallelogram

Considering a two-link two-DOF mechanism, there are

some methods to balance the gravity of link mass shown in

Fig. 2. The gravitational potential energy of total masses is

computed by

𝑈𝑚 = (𝑚1𝑔𝑙1 + 𝑚2𝑔𝐿1) cos 𝜃1

+𝑚2𝑔𝑙2 cos(𝜃1 + 𝜃2)(4)

Here 𝑚𝑖 is the mass of link 𝑖, 𝑙𝑖 is the length of the weight

arm of link 𝑖, 𝜃𝑖 is the rotational angle of the joint 𝑖. 𝐿1 is

the length of link 1. To balance the gravity, the system

needs the counterbalancing springs with the potential

energy that varies in the form, cos 𝜃1 and cos(𝜃1 + 𝜃2).

Therefore the cos 𝜃1 denotes a spring balancer with the

transmission related to joint 1. And the cos(𝜃1 + 𝜃2)denotes a spring balancer with the transmission related to

joint 1 and joint 2. The parallelogram and cable driven as

the transmission can easily satisfy the function, 𝜃1 + 𝜃2.

Although there are many types for one certain mechanism

due to the variety of one-DOF spring balancer and the

transmission, the each spring finally has the same

expression, just as

𝑈𝑠1= 𝐶 − 𝑚𝑔𝑙 𝑐𝑜𝑠 𝜃1 (5)

𝑈𝑠2= 𝐶 − 𝑚𝑔𝑙 𝑐𝑜𝑠(𝜃1 + 𝜃2) (6)

Fig. 3. (a) One-link roll-pitch manipulator (b) Gravity compensation using the bevel gear in [12]

A one-link roll–pitch manipulator is depicted in Fig. 3.

Respectively, the gravitational potential energy of the

masses is computed by

𝑈𝑚 = 𝑚1𝑔𝑙1 cos 𝜃1 cos 𝜃2 (7)

The cos 𝜃1 cos 𝜃2 cannot directly match the structure

made up of the spring balancer and transmission. It was

always transformed through the product-to-sum identity

cos 𝜃1 cos 𝜃2 = 0.5 cos(𝜃1 + 𝜃2) + 0.5 cos(𝜃1 − 𝜃2) (8)

This equation represents the transmission is similar to a

mechanical differential to satisfy the both 𝜃1 + 𝜃2 and

𝜃1 − 𝜃2 . The mechanical differential, the gear or cable

driven differential, connects with the two spring balancers

to compensate the gravity. These springs generate

correspondent elastic potential energy

𝑈𝑠1= 𝐶 − 0.5𝑚1𝑔𝑙1 cos(𝜃1 + 𝜃2) (9)

𝑈𝑠2= 𝐶 − 0.5𝑚1𝑔𝑙1 cos(𝜃1 − 𝜃2) (10)

As a summary, these calculations show that multi-link

and multi-DOF spring gravity compensations are the

system of several spring balancers and transmission. The

mathematic function of these transmissions is the form of

the linear sum of each joint angle. This implies that the

gravitational potential energy should be divided into

different energy bases according to the transmission

structure. That is to say, the transmission decides the

method of multi-link and multi-DOF spring gravity

compensation. To obtain a new method, one way is to

change the transmission followed by the change of its

expression.

III. Spring compensation with nonlinear transmission

This section introduces the transmission using the

noncircular pulley which is defined as nonlinear

transmission. Spring balancer with a nonlinear

transmission provides an elastic potential energy of new

expression. The method of fabricating the possible

nonlinear transmission for gravity compensation is

proposed.

A. Nonlinear transmission

To adjust the mapping between the input and output,

special transmission is used in mechanism. I. Simionescu

[3, 4] developed the discrete and continuous transmission

using the multi-link and noncircular pulley for gravity

compensation. The aim of these designs is to modify the

motion of springs to match the motion of the link. The

noncircular pulley transmission has more freedom than

multi-link transmission to generate the functional potential

energy and achieve the perfect balancing. Nonlinear

transmission can be presented by replacing the circular

pulley with the noncircular pulley. As shown in Fig. 4, the

mathematics expression of nonlinear transmission is

discussed below.

Fig. 4. Cable driven using the noncircular pulley

The Fig. 4 shows a serial of cable transmission structure.

With respect to the every figure, the relationship of the

rotation angle between the first pulley and links can be

described as follows

𝜃0𝑝

= 𝑓1(𝜃1) (11)

𝜃1𝑝

= 𝜃1 + 𝑓2(𝜃2) (12)

𝜃1𝑝

= 𝜃1 + 𝑓2(𝜃2 + 𝑓3(𝜃3 + ⋯ )) (13)

Here 𝜃𝑖𝑝

denotes the rotation angle of the pulley i, 𝜃𝑖

denotes the rotation angle of the link i, the mapping 𝑓𝑖

presents the rotation angle of the pulley 𝑖 − 1 caused by

the rotation angle of the pulley 𝑖. In Fig. 4(a), the rotation

of pulley 0 is only caused by the rotation of the pulley 1

which is the same as the rotation of the link 1. In Fig. 4(b),

the rotation of pulley 1 is caused by the rotation of pulley 2

and the rotation of link 1. It can be expressed as 𝜃1𝑝

= 𝜃1 +

𝑓2(𝜃2𝑝

) where 𝜃2𝑝

= 𝜃2. It can be expanded into the general

expression as 𝜃𝑖−1𝑝

= 𝜃𝑖 + 𝑓𝑖(𝜃𝑖−1𝑝

). By combining all the

equation of the rotation angles of each adjacent pulley, Eq.

(13) can be concluded as the general expression for

nonlinear transmission.

B. Potential energy with nonlinear transmission

The energy base which presents the gravitational

potential energy balanced by one spring is determined by

transmission structure. To distinguish the two kinds of the

energy base, the energy base decided by the linear

transmission is defined as the linear energy base. Relatively,

the energy base decided by the nonlinear transmission is

defined as the nonlinear energy base.

Fig. 5. One-DOF spring balancer

connected with cable driven transmission

Consider that two kinds of the transmission connect with

the one-DOF spring balancer as shown in Fig. 5. The

general expression of potential energy base can be

expressed as

𝑈𝑗𝑙 = cos (𝜃1 ± ⋯ ±𝜃𝑛) (14)

𝑈𝑗𝑛𝑙 = cos (𝜃1 ± 𝑓2𝑗 (𝜃2 ± 𝑓3𝑗(𝜃3 ± ⋯ ))) (15)

Here 𝑈𝑗𝑙 denotes the linear energy base balanced by the

spring 𝑗, 𝑈𝑗𝑛𝑙 denotes the nonlinear energy base balanced

by the spring 𝑗 , 𝑓𝑛𝑗 denotes the rotation mapping in

transmission connected with spring 𝑗.

Notice that the mapping 𝑓𝑛𝑗 is developed by the

requirement, 𝑈𝑗𝑛𝑙 presents a more variable function than 𝑈𝑗

𝑙.

When the mapping 𝑓𝑛𝑗 is equal to 1, 𝑈𝑗𝑛𝑙 and 𝑈𝑗

𝑙 are same.

C. Methodology for the transmission function

Although there are two kinds of energy bases, the sum of

linear or nonlinear energy bases must be equal to the entire

gravitational potential energy for one statically balancing

mechanism. This implies that the sum of some linear

energy bases must be equal to the sum of some nonlinear

energy bases. This relationship can be described as

𝑉𝑚 = ∑ 𝑈𝑗𝑙 = ∑ 𝑈𝑗

𝑛𝑙 (16)

The question to obtain the category of nonlinear

transmission is equivalent to the combination category of

the nonlinear energy bases. Considering a nonlinear energy

base with two pulleys in a general situation, it can be

expressed as

𝑈𝑖𝑛𝑙 = cos(𝜃1 + 𝑓𝑖(𝜃2)) (17)

𝑈𝑖𝑛𝑙 = cos 𝜃1 cos(𝑓𝑖(𝜃2)) − sin 𝜃1 sin(𝑓𝑖(𝜃2)) (18)

We assume that the mapping 𝑓𝑖 is not equal to 1. Notice

that the nonlinear energy base cannot be independently

equal to any one linear energy base, just as cos (𝜃1 +𝜃2), cos (𝜃1 − 𝜃2), cos (𝜃1). The sum of any two nonlinear

energy bases with two pulleys can be expressed as

𝐾1𝑛𝑙𝑈1

𝑛𝑙 + 𝐾2𝑛𝑙𝑈2

𝑛𝑙 =

cos 𝜃1 𝐹1𝑛𝑙(𝜃2) − sin 𝜃1 𝐹2

𝑛𝑙(𝜃2)(19)

where

𝐹1𝑛𝑙(𝜃2) = 𝐾1

𝑛𝑙 cos(𝑓1(𝜃2)) + 𝐾2𝑛𝑙 cos(𝑓2(𝜃2)) (20)

𝐹2𝑛𝑙(𝜃2) = 𝐾1

𝑛𝑙 sin(𝑓1(𝜃2)) + 𝐾2𝑛𝑙 sin(𝑓2(𝜃2)) (21)

Here 𝐾𝑖𝑛𝑙 denotes the parameter of one nonlinear energy

base. Eq. (26) and Eq. (27) present two equations with two

undetermined mapping 𝑓21 and 𝑓22 . This implies 𝐹1(𝜃2)and 𝐹2(𝜃2) has more freedom to match the expression of

the entire potential energy. Then the sum of two linear

energy bases, 𝑈1𝑙 = cos (𝜃1 + 𝜃2) and 𝑈2

𝑙 = cos (𝜃1 − 𝜃2)can be expressed as

𝐾1𝑙𝑈1

𝑙 + 𝐾2𝑙𝑈2

𝑙 =

cos 𝜃1 𝐹1𝑙(𝜃2) − sin 𝜃1 𝐹2

𝑙(𝜃2)(22)

where

𝐹1𝑙(𝜃2) = 𝐾1

𝑙 cos(𝜃2) + 𝐾2𝑙 cos(𝜃2) (23)

𝐹2𝑙(𝜃2) = 𝐾1

𝑙 sin(𝜃2) − 𝐾2𝑙 sin(𝜃2) (24)

Here 𝐾𝑖𝑙 denotes the parameter of one linear energy base

and 𝐾𝑖𝑙 is known decided by the mechanism. According to

the relationship of the elastic potential energy for one

certain mechanism, Eq. (16) requires that the 𝐾1𝑛𝑙𝑈1

𝑛𝑙 +𝐾2

𝑛𝑙𝑈2𝑛𝑙 is equal to the 𝐾1

𝑙𝑈1𝑙 + 𝐾2

𝑙𝑈2𝑙 . Therefore the

relationship between 𝐹𝑖𝑛𝑙(𝜃2) and 𝐹𝑖

𝑙(𝜃2) is expressed as

𝐹1𝑛𝑙(𝜃2) = 𝐹1

𝑙(𝜃2) (25)

𝐹2𝑛𝑙(𝜃2) = 𝐹2

𝑙(𝜃2) (26)

The Eq. (25) and Eq. (26), two nonlinear equations with

respect to the 𝜃2, determine the two unknown mappings

and two unknown parameter 𝐾𝑖𝑛𝑙. It is obvious that the

four unknown quantity can be calculated.

The following contents describe some cases to achieve

the transformation.

Fig. 6. Multi-DOF or Multi-link manipulator

Case I:

The nonlinear energy base, such as cos(𝑓1(𝜃1)), is a

function about single joint angle 𝜃1 . For some 1-DOF

multi-link mechanism, the entire gravitational potential

energy cannot be simplified to the form as 𝑐𝑜𝑠 𝜃. Cho [15]

presents a design method for static balancer with the

associated linkage. In his reach, various gravity

compensations are designed for four-bar linkage and slider

crank. Several springs and parallelogram are adopted, but

spring numbers are more than their DOFs.

For 1-Dof close-loop mechanism to reduce spring

number, we assume all the gravitational torques caused by

linkage mass effect on the single joint. Because the other

joint motion is correlated with this single joint, the

potential energy can be expanded as one variable function.

As shown in Fig. 6(a), it is described as

𝑉𝑚 = 𝐴 cos 𝜃1 + 𝐵 cos(𝜃1 + 𝜃2)

+𝐶 cos(𝜃1 + 𝜃2 + 𝜃3)(27)

where

𝐴 = 𝑚1𝑔𝑙1 + (𝑚2 + 𝑚3)𝑔𝐿1

𝐵 = 𝑚2𝑔𝑙2 + 𝑚3𝑔𝐿2

𝐶 = 𝑚3𝑔𝑙3

(28)

Notice that the angle 𝜃2 and 𝜃3 is related to 𝜃1. The Eq. (27)

can be written as

𝑉𝑚 = 𝐴 cos 𝜃1 + 𝐵 cos(𝑓𝑎(𝜃1)) + 𝐶 cos(𝑓𝑏(𝜃1)) (29)

where

𝑓𝑎(𝜃1) = 𝜃1 + 𝜃2

𝑓𝑏(𝜃1) = 𝜃1 + 𝜃2 + 𝜃3

(30)

Here 𝑓𝑎 and 𝑓𝑏 denote the mapping between these angles.

Furthermore the Eq. (29) is described as the desired form

which represents the nonlinear energy base as

𝑉𝑚 = 𝐾𝑛𝑙 cos(𝑓𝑠(𝜃1)) (31)

The Eq. (31) shows that one-DOF four-bar mechanism

can be balanced by one nonlinear 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 also

designed using the nonlinear transmission for multi-Dof

mechanism.

As shown in Fig. 6(b), the potential energy can be

described as

𝑉𝑚 = 𝐴 cos 𝜃1 + 𝐵 cos(𝜃1 + 𝜃2)

+𝐶 cos(𝜃1 + 𝜃2 + 𝜃3)

+𝐷 cos(𝜃1 + 𝜃2 + 𝜃3 + 𝜃4)

(32)

where

𝐴 = 𝑚1𝑔𝑙1 + (𝑚2 + 𝑚3 + 𝑚4)𝑔𝐿1

𝐵 = 𝑚2𝑔𝑙2 + 𝑚3𝑔𝐿2 + 𝑚4𝑔𝐿2

𝐶 = 𝑚3𝑔𝑙3 + 𝑚4𝑔𝐿3

𝐷 = 𝑚4𝑔𝑙4

(33)

Notice that the angle 𝜃3 and 𝜃4 is only related to 𝜃2. The

Eq. (32) can be also written as

𝑉𝑚 = 𝐴 cos 𝜃1 + 𝐵 cos(𝜃1 + 𝜃2)

+𝐶 cos(𝜃1 + 𝑓𝑎(𝜃2)) + 𝐷 cos(𝜃1 + 𝑓𝑏(𝜃2))(34)

where

𝑓𝑎(𝜃2) = 𝜃2 + 𝜃3

𝑓𝑏(𝜃2) = 𝜃2 + 𝜃3 + 𝜃4

(35)

The Eq. (31) shows that the potential energy of the entire

mechanism is the function with respect to 𝜃1 and 𝜃2 .

According to the method from Eq. (16) to Eq. (23), the Eq.

(34) can be written as

𝑉𝑚 = 𝐾1𝑛𝑙 cos(𝜃1 + 𝑓1(𝜃2))

+𝐾2𝑛𝑙 cos(𝜃1 + 𝑓2(𝜃2))

(36)

The Eq. (36) shows that mechanism depicted in Fig. 6(b)

can be balanced by two nonlinear transmissions and two

springs.

Case III:

The mechanism shown in Fig. 6(c) is similar to the

mechanism shown in Fig. 6(b). And the potential energy

can be described as

𝑉𝑚 = 𝐴 cos 𝜃1 cos 𝜃2 + 𝐵 cos 𝜃1 cos (𝜃2 + 𝜃3)

+𝐶 cos 𝜃1 cos (𝜃2 + 𝜃3 + 𝜃4)(37)

where

𝐴 = 𝑚1𝑔𝑙1 + (𝑚2 + 𝑚3)𝑔𝐿1

𝐵 = 𝑚2𝑔𝑙2 + 𝑚3𝑔𝐿2

𝐶 = 𝑚3𝑔𝑙3

(38)

Notice that the angle 𝜃3 and 𝜃4 is only related to 𝜃2. Using

the same transforming above, the Eq. (37) can be also

written as

𝑉𝑚 = 𝐴 cos 𝜃1 cos 𝜃2 + 𝐵 cos 𝜃1 cos (𝑓𝑎(𝜃2))

+𝐶 cos 𝜃1 cos(𝑓𝑏(𝜃2))(39)

where

𝑓𝑎(𝜃2) = 𝜃2 + 𝜃3

𝑓𝑏(𝜃2) = 𝜃2 + 𝜃3 + 𝜃4

(40)

Furthermore the Eq. (39) is expressed as

𝑉𝑚 = 𝐾𝑛𝑙 cos(𝜃1 + 𝑓(𝜃2))

+𝐾𝑛𝑙 cos(𝜃1 − 𝑓(𝜃2))(41)

where

2𝐾𝑛𝑙 cos(𝑓(𝜃2)) = 𝐴 cos 𝜃2 + 𝐵 cos(𝑓𝑎(𝜃2))

+𝐶 cos(𝑓𝑏(𝜃2))(42)

Case IV:

As shown in Fig. 6(d), the expression of gravitational

potential energy of the two-DOF pitch-yaw mechanism can

be expanded as

𝑉𝑚 = 𝐴 cos 𝜃1 + 𝐵 cos 𝜃1 cos 𝜃2 (43)

where

𝐴 = 𝑚1𝑔𝑙1 + 𝑚2𝑔𝐿

𝐵 = 𝑚2𝑔𝑙2

(44)

Furthermore the Eq. (43) is extended as

𝑉𝑚 = 𝐾𝑛𝑙 cos(𝜃1 + 𝑓(𝜃2))

+𝐾𝑛𝑙 cos(𝜃1 − 𝑓(𝜃2))(45)

where

2𝐾𝑛𝑙 cos(𝑓(𝜃2)) = 𝐴 + 𝐵 cos(𝜃2) (46)

As a summary, the gravity compensation using the

nonlinear transmission needs less spring for some special

mechanisms. However this method cannot be used in all

the mechanism to reduce spring number for perfect gravity

compensation. It depends on the transformation of the

expression of gravitational potential energy. Some items,

just as constant value or one variable function, can be

combined together into one item. The design to realize this

mathematic procedure is to change the transmission

mapping and the curve of the pulley.

IV. Synthesis of the noncircular pulley

A. Calculation for the curve of noncircular pulley

The transmission mapping 𝑓 is determined by the

procedure mentioned in above chapter. The geometry

curve of the noncircular pulley depends on the certain

mapping 𝑓 . Due to nonlinearity of the transmission

mapping, the synthesis of noncircular pulley is based on

the infinitesimal calculus approach which leads to an

analytical solution. With the help of former study [3], there

needs a series of calculations step by step: 1. the length of

the moment arm, 2. the position of the cable, 3. the

geometry curve of the noncircular pulley. In this paper the

moment arm is obtained through the transmission

mapping 𝑓.

Fig. 7. A schematic diagram of the transmission

with the noncircular pulley and cable

The first step is to derive the relationship between the

moment arms of two adjacent pulleys by using the

principle of virtual work. As shown in Fig. 7(a), the pulley

𝑎 with driving torque 𝜏𝑎 drives the pulley 𝑏 with the load

torque 𝜏𝑏 in a constant speed by cable driven. So the virtual

work of the system and the relationship between the

motions of two pulleys can be expressed as

𝜏𝑎𝛿𝜃𝑎 = 𝜏𝑏𝛿𝜃𝑏 (47)

𝑓(𝜃𝑎) = 𝜃𝑏 (48)

𝑓′(𝜃𝑎)�̇�𝑎 = �̇�𝑏 (49)

Because the torque is equal to the product of the moment

arm and the cable tension, it is described as

𝜏𝑎 = 𝐹𝑐𝑎𝑏𝑙𝑒𝑟𝑎(𝜃𝑎) (50)

𝜏𝑏 = 𝐹𝑐𝑎𝑏𝑙𝑒𝑟𝑏(𝜃𝑎) (51)

Here, 𝐹𝑐𝑎𝑏𝑙𝑒 denotes the tension in cable, 𝑟𝑖(𝜃𝑎) denotes

the moment arm of the pulley 𝑖 with angle variable 𝜃𝑎 .

Considering 𝛿𝜃 is equal to �̇�, the overall procedure derives

the relationship between the transmission mapping and the

radius of the pulley.

𝑟𝑎(𝜃𝑎) 𝑟𝑏(𝜃𝑎)⁄ = 𝑓′(𝜃𝑎) (52)

Fig. 8. A schematic diagram of the geometry curve

of the noncircular pulley

The second step is to obtain the planar position of the

cable on one pulley with respect to all configurations. The

radius of pulley 𝑎 is determined by 𝑓′(𝜃𝑎) and 𝑟𝑏(𝜃𝑎) at

each angle of the pulley 𝑎. For simplification the pulley 𝑏

is considered as circle pulley, so that the radius and the

moment arm are constant. As shown in Fig. 7(a), the

equation of the line 𝐴𝐵 which presents the cable’s position

at the joint angle of 𝜃𝑎 can be expressed as

𝑦 = 𝐴(𝜃)𝑥 + 𝐵(𝜃) (53)

where

𝐴(𝜃) = tan(𝜃 + 𝜇) (54)

𝐵(𝜃) = 𝑟𝑎(cos(𝜃 + 𝜇) − sin(𝜃 + 𝜇) tan(𝜃 + 𝜇)) (55)

𝜇 = arcsin ((𝑟𝑎(𝜃) − 𝑟𝑏(𝜃)) 𝑙𝑜⁄ ) (56)

Here 𝑙𝑜 denotes the distance between the two pulleys.

The third step is to obtain the equations of the geometry

curve. Once the moment arms of each pulley are known at

the entire possible angles, the planar positions of the

transmission cable trance out the profile of the pulley. As

shown in Fig. 7(b), the position of the transmission cable

changes with respect to the motion of pulley 𝑎. When the 𝜃

changes infinitesimally, the intersection point P of the two

lines at joint angle 𝜃 and 𝜃′ is at the profile of the pulley.

So the position of the point P can expressed as

𝑥𝑝 = − (𝐵(𝜃′) − 𝐵(𝜃)) (𝐴(𝜃′) − 𝐴(𝜃))⁄ (57)

𝑦𝑝 = (𝐴(𝜃′)𝐵(𝜃) − 𝐴(𝜃)𝐵(𝜃′)) (𝐴(𝜃′) − 𝐴(𝜃))⁄ (58)

where 𝜃′ = 𝜃 + 𝛿𝜃, 𝑥𝑝 and 𝑦𝑝 are the coordinates of the

intersection point 𝑃. When the 𝛿𝜃 becomes infinitesimal,

Eq. (57) and Eq. (58) can be expanded as

𝑥𝑝(𝜃) = − 𝐵′(𝜃) 𝐴′(𝜃)⁄ (59)

𝑦𝑝(𝜃) = − 𝐴(𝜃)𝐵′(𝜃) 𝐴′(𝜃)⁄ + 𝐵(𝜃) (60)

B. Parameter selection

It is to be necessary that the profile of the pulley should

be a continuous differentiable curve to reach the proper

transmission. In the research [16], it decided that the

transmission mapping cannot be all the required functions.

Once the derivation of the transmission mapping, 𝑓′(𝜃),

changes sign, the curve is a strong nonlinear that leads the

interference of the cable. To avoid this situation, the

transmission mapping should be selected to generate the

proper profile of the pulley.

Fig. 9. Geometry curve of the noncircular pulley

with different parameter 𝐾

𝑚1 𝑚2 𝑙1 𝑙2 𝐿1

0.3kg 0.5kg 0.2m 0.2m 0.3m

𝑟𝑏 𝑙𝑜 𝑔 𝜃2

10cm 30cm 0.98 [−135o, 135o]

Table. 1. The design parameters

For example, the transmission function of Case IV in

Section III is 𝐾 cos(𝑓(𝜃2)) = 𝑚1𝑔𝑙1 + 𝑚2𝑔𝐿 +

𝑚2𝑔𝑙2 cos 𝜃2. The value of the parameter 𝐾 chosen in a

range decides both the geometry curve of the noncircular

pulley and motion range of the joint.

As shown in Fig. 9, three geometry curves are

demonstrated with the parameters in Table. 1. Consider

that 𝑇 = 𝑚1𝑔𝑙1 + 𝑚2𝑔𝐿 + 𝑚2𝑔𝑙2. When the value of 𝐾is equal to 𝑇, the curve is a continuous line and the motion

range is wide enough from −π to π. When the 𝐾 is less or

great than 𝑇, the curve is discrete and the motion range is

decreasing.

C. The design of the antagonistic cable driven

In many robots, the gravitational torque is bi-directional

effect on the joint. To obtain more motion space with the

configuration of the gravity compensation, the antagonistic

cable driven design is necessary to transmit bi-directional

torque.

According to close-loop transmission of the cable driven

using the circle pulley, we consider that there are two lines

on one pulley which should be calculated separately. When

the cable wraps a distance around the pulley on the one side,

the cable unwraps the same distance on the other side. The

antagonistic design of the noncircular pulley is to maintain

the length of each moment arm equal on each side.

Fig. 10. The antagonistic design for cable driven

Different from the circle pulley, two lines of the

noncircular pulley cannot connect with each other by end

to end in most cases. As shown in Fig. 10, the profile of the

noncircular pulley consists of two lines separately which

overlaps each other. Notice that the geometry of two lines

is the same located at different angles. It leads to that the

noncircular pulley can be assembled with two same pulleys

in two layers.

V. Design examples

This section presents a design example of spring gravity

compensation for two-DOF pitch-yaw manipulator. The

new design method using the noncircular pulley will

reduce the number of the springs and abbreviate the entire

structure. The transmission structure and the profile of the

each pulley are discussed and calculated in more details.

Fig. 11. Two-DOF pitch-yaw manipulator

The manipulator is depicted in Fig. 11. 𝜃1 and 𝜃2

represent rotation angles in the 𝑧1 and 𝑧2 axes, respectively.

The gravitational potential energy is computed by

𝑉𝑚 = 𝑉𝑚1+ 𝑉𝑚2

𝑉𝑚1= 𝑚1𝑔𝑙1 cos 𝜃1

𝑉𝑚2= 𝑚2𝑔𝑙2 cos 𝜃1 cos 𝜃2 + 𝑚2𝑔𝐿1 cos 𝜃1

where 𝑙𝑖 denotes the moment arm of the link 𝑖, 𝐿𝑖 denotes

the distance of the link. Notice that the entire potential

energy has two energy items of 𝑐𝑜𝑠 𝜃1 𝑐𝑜𝑠 𝜃2 and 𝑐𝑜𝑠 𝜃1

which can generate three linear energy bases,

𝑈1𝑙 = 𝐾1

𝑙 cos 𝜃1

𝑈2𝑙 = 𝐾2

𝑙 cos(𝜃1 + 𝜃2)

𝑈3𝑙 = 𝐾3

𝑙 cos(𝜃1 − 𝜃2)

where 𝐾𝑖𝑙 denotes the parameter of the linear energy base 𝑖.

So 𝐾1𝑙 = 𝑚1𝑔𝑙1 + 𝑚2𝑔𝐿1 , 𝐾2

𝑙 = 𝐾3𝑙 = 0.5𝑚2𝑔𝑙2 . Three

linear energy bases denote the gravity compensation of this

manipulator should have three transmission chains and

three springs using the circle pulley or parallelogram.

According to the method described in Section III, two

nonlinear energy bases, instead of three linear energy bases,

can be fabricated to reduce the number of springs and

cables. For simplicity, the specific expanding is

demonstrated as follows

𝑉𝑚 = 𝐾 cos 𝜃1 cos(𝑓(𝜃2))

𝐾 cos(𝑓(𝜃2)) = 𝑚2𝑔𝑙2 cos 𝜃2 + 𝑚2𝑔𝐿1 + 𝑚1𝑔𝑙1

Then, the spring compensation system consists of two

nonlinear energy bases, just as

𝑈1𝑛𝑙 = 0.5𝐾1

𝑛𝑙 cos(𝜃1 + 𝑓(𝜃2))

𝑈2𝑛𝑙 = 0.5𝐾2

𝑛𝑙 cos(𝜃1 − 𝑓(𝜃2))

where 𝐾𝑖𝑛𝑙 denotes the parameter of the nonlinear energy

base 𝑖, 𝐾1𝑛𝑙 = 𝐾2

𝑛𝑙 = 𝐾.

(a)

(b) Fig. 12. Spring gravity compensation for two-DOF pitch-yaw

mechanism: (a) planar schematic (b)stereoscopic model

The relevant parameters for the design are chosen as:

𝑚1 = 0.3kg , 𝑚2 = 0.5kg , 𝐿1 = 0.3𝑚 , 𝑙1 = 𝑙2 = 0.2𝑚 ,

𝑟1 = 0.1𝑚. And 𝐾 is calculated to be 0.1514N. m. In Fig.

12, the entire part of the transmission and mechanism is

demonstrated by planar schematic and stereoscopic model.

Pose 𝜃1 𝜃1

1 0° 0°

2 0° 15°

3 15° 0°

4 15° 15°

5 15° 30°

6 30° 15°

7 30° 30°

Table. 2. Poses for simulation

Fig. 13. Computation of the potential energy

To verify the effectiveness of the method, seven poses

are chosen to calculate the elastic potential energy and

gravitational potential energy of the mechanism. The pose

index is shown in Table. 2. As shown in Fig. 17, the total

potential energy maintains the invariant for all the poses.

VI. Conclusion of the present work

This paper presents a new method for static balancing of

mechanisms with conservative loads such as gravity and

spring loads using noncircular pulley and cable driven. The

method, which completely balances gravitational torque,

provides reduction the actuator requirements under space

constraints.

The transmission structure is key factor to determine the

expression of the potential energy of the springs. The new

transmission structure, using the noncircular pulley at the

start of the transmission, is adopted to generate more

flexible spring force for gravity compensation. The entire

gravitational potential energy is divided into several parts

which correspond with the new structure. This paper uses a

simple way, transformation of the expression of the

potential energy, to find out the transmission structure and

coefficient of the spring. As a result, the number of the

springs is reduced for the gravity compensation of some

special mechanism.

Although the method cannot apply in all the mechanism

to reduce the spring number, it is a new way to achieve a

complete static balancing. With no auxiliary linkages and

suspending cable, the resultant mechanism is more

compact.

We implement the theory of the design of the noncircular

pulley based on the former study. According to the function

of the transmission mapping, the geometry curve of the

noncircular pulley is calculated using the infinitesimal

mathematic process. The circle pulley and noncircular

pulley are arranged in a serial by cable driven. The

selection and a needed structure of antagonistic cable

driven are demonstrated.

Future work is to find out the more availability of the

transmission structures which present more availability of

the expression of the potential energy. We believe the more

mechanism adopt flexible transmission to have a less

spring number structure for gravity compensation.

Acknowledgment

Research supported by National Natural Science

Foundation of China under No. 51290293.

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