Abstract—High mobility, maneuverability and obstacle surmounting capabilities are highly desirable features for rough-terrain locomotion systems. In past work, we examined kinetostatic optimization of candidate articulated leg-wheel subsystem designs (based on the four-bar mechanism) for enhancing locomotion capabilities of land-based vehicles. Our goal was to: (i) achieve the greatest motion-ranges between wheel axle and chassis while (ii) reducing the overall actuation requirements by spring assist. In the current work, we examine the possibility of enhancing this terrain-accommodation by: (i) proposing an “adjustable four-bar” articulated-leg-wheel subsystem; (ii) with active-structural control to actively change subsystem parameters during the terrain traversal. Multiple leg-wheel design- parameters can affect the peak-static torque requirements as well as dynamic-bandwidth requirements for the leg-wheel actuation. The presented results compare and contrast online-structural reconfiguration planning (with multiple alternate active-adjustments) to reduce actuation requirements for a predetermined/sensed terrain traversal profile.

I. INTRODUCTION

In recent years, land-based locomotion systems for operation on unstructured terrain have generated considerable interest among researchers with increasingly diverse applications such as rough-terrain planetary explorations and disaster environments [1].

Hybrid articulated leg-wheel subsystems (of the type shown in Fig.1) are often considered in order to exploit the benefits of both legged [2, 3] and wheeled systems [4] alternate articulated leg-wheel designs (with multiple revolute/prismatic joints between the wheel and the chassis) could be realized depending upon the type, number, sequencing of the joints. Adding additional articulations increases the intermediate degree of freedom within the chain, which need to be controlled or restricted either: (i) actively by actuation, (ii) semi-actively using springs and dampers; or (iii) passively by adding some form of structural equilibration using hardware constraints. Well-studied planetary rovers with leg-wheel subsystem range from the Sojourner [5], Rocky 7 [6], Micro5 [7], SHRIMP [8] with rocker bogie suspensions, the NOMAD [9], and other

This work is partially supported by NSF Grant IIS-1319084 and CNS-1314484.

*Aliakbar Alamdari, Javad Sovizi and Seung Kook Jun are PhD Candidates with Department of Mechanical and Aerospace Engineering, the State University of New York at Buffalo, NY 14260 USA (phone: 716-480-6048; e-mail: {aalamdar, javadsov, seungjun}@ buffalo.edu).

Venkat N. Krovi is an Associate Professor in Department of Mechanical and Aerospace Engineering, the State University of New York at Buffalo, (e-mail: vkrovi@buffalo.edu).

wheeled actively articulated vehicle WAAV [10,15,16] designs with articulated frames which can typically overcome obstacles of their wheel size, if friction is high enough, to systems like the Work Partner [11], Roller-Walker [12], and ALDURO [13] with powered legs and active/passive wheels which mostly utilize special strategy and often complex control procedure.

In addition to foremost necessities (such as high mobility, obstacle-surmounting capability and maneuverability for operation on unprepared surfaces), there are additional criteria (such as robustness, reliability, and efficiency) to be considered in the design process. In almost all cases no systematic effort to “design” the articulated leg-wheel system is ever considered. In past work [14], we examined candidate leg-wheel subsystems and explored the development of an optimization framework.

Figure 1. Candidate design for adjustable articulated leg-wheel

subsystem In the current work, we extend this framework for

enhanced terrain accommodation by: (i) proposing an “adjustable four-bar” articulated leg-wheel subsystem; (ii) with active structural parameters control to change subsystem parameters during motions (as shown in Fig.1). Three design variants (with adjustable crank, coupler, or follower) will be created and parameters will be optimized using design optimization framework. Since the load-bearing requirements can be significant and are highly configuration dependent, we will focus on exploiting structural equilibration to bear these loads. In this paper, we will enhance the kinetostatic optimization to allow for online-structural reconfiguration planning. Multiple staged active-adjustments of the structural-configuration will be studied to reduce actuation requirements for traversal profile in an online manner.

The rest of this paper is organized as follows: In Section 2, we provide a brief background of the general kinematic and kinetostatic design approaches used for mechanism design. In section 3, we specialize the kinematic/kinetostatic approach for the adjustable four-bar leg-wheel subsystem to: (i) surmount varying step profiles, and (ii) reduce actuation

Kinetostatic Optimization for an Adjustable Four-Bar Based Articulated Leg-Wheel Subsystem

Aliakbar Alamdari*, Javad Sovizi, Seung Kook Jun and Venkat N. Krovi

2014 IEEE/RSJ International Conference onIntelligent Robots and Systems (IROS 2014)September 14-18, 2014, Chicago, IL, USA

978-1-4799-6933-3/14/$31.00 ©2014 IEEE 2860

requirements. In Section 4, alternate design variants for realizing adjustment capability will be presented and Section 5 presents concluding remarks.

II. BACKGROUND

A. Kinematic Synthesis Dimensional/ Kinematic synthesis is the process of

generating the geometry of the mechanism that will perform a specific task. In here, geometry generation means determining the lengths of the individual links that make up the mechanism to perform a desired function generation, path generation, and motion generation [17]. For all these three synthesis types, the prescribed conditions that the mechanism must satisfy are called the “precision positions”. The resulting linkage can create the desired motion precisely at precision positions and approximately at other positions. The more precision points are used, the closer is the actual motion of the coupler to the ideal desired motion. But the problem becomes more difficult to solve as the number of precision positions is increased. Fortunately, many real world problems only need few critical positions to be satisfied precisely. A four-bar linkage can satisfy up to five prescribed positions for the motion generation problem. The adjustable four-bar linkage, on the other hand, can satisfy more than five given positions by making some of the parameters adjustable.

Adjustable mechanisms, sometimes called reconfigurable or programmable mechanisms are capable of generating multiple paths by changing link length or fixed pivot positions. One of the first approaches towards synthesis of adjustable four-bar mechanisms was using the well-known complex number method. An analytical method with closed-form solution and utilizing complex numbers to synthesize an adjustable four-bar path generator to generate two different paths was suggested by McGovern and Sandor [19]. Chuenchom and Kota [18] applied analytical synthesis method for adjustable link lengths and ground pivots. The actual mathematical procedure is involving solution of six-degree polynomials. In recent works on adjustable four-bar mechanisms, a two-stage design method was proposed [20, 21]. In the first stage, the driving dyad is determined and in the second stage the driven dyad is obtained. Both design stages used sequential quadratic optimization (SQP) algorithm to search for the optimal design variables. In [21], they used a least-squares based circle fitting procedure in the second stage which results in less number of search variables which in turn results in more efficient optimization. The objective functions of both stages are based on the geometry of the adjustable four-bar mechanism.

Optimal synthesis methods minimize an error function of the difference between the generated and the desired motions, called the Structural Error, to find the mechanism that gives the best approximation of the desired motion. For path generation, structural error has been conventionally defined as the sum of squares of the distances between the desired and the generated path points.

B. Kinetostatic Synthesis One of the main issues which have the most effect on the

performance of leg-wheel design is the force interaction between a linkage and its environment. The kinetostatic synthesis is used to assist the selection of the optimal configuration (and determine the minimum torques required) to support these external loads by structural equilibration. The goal of the articulation is to guide the attached wheel through several positions while supporting a set of specified external loads. We will also add torsional springs at each joint to minimize/optimize the peak actuator torques by a sensible combination of structural equilibration design and spring assists. The formulation and simultaneous solution of the loop-closure and static equilibrium equations at the precision points aids the process of simultaneous determination of optimal mechanism and spring parameters [14].

III. DESIGN PROCESS FOR ADJUSTABLE LEG-WHEEL SUBSYSTEM

Precision Point Synthesis (PPS) is coupled with optimization to realize a desired path for the wheel axle of the adjustable four-bar leg-wheel subsystem. Emphasis is placed both on satisfying the kinematic and static specifications in both exactly at precision points in the least squares sense elsewhere.

A. Position Analysis (Finite and Differential) The finite loop closure equation of the adjustable four-bar

leg-wheel subsystem (Fig.2) can be written as:

Figure 2. Adjustable four-bar mechanism

  (1)

The dependent virtual angular displacements of and can be expressed in terms of the independent virtual displacement of and as:

 (2)

B. Kinematic Synthesis The calculation of geometric parameters of articulated

mechanical or robotic system so that it passes a link in a number of specified successive location or precision points in the plane is referred as Geometric Design Problem [17].

2 2 3 3 4 4

2 2 3 3 1 4 4

Kz cos z cos z cosKz sin z sin z z sin

θ + θ = θθ + θ = − + θ

θ3(δγ )

θ4(δβ)

θ2(δα) K(δk)1

3 3 4 4

3 3 4 4

2 2 2 2 11 12

2 2 2 2 21 22

z sin( ) z sin( )z cos( ) z cos( )

Kz sin( ) z cos( ) C CKz cos( ) z sin( ) C CK K

−Θ +β − Θ + γδβ ⎡ ⎤⎡ ⎤= ⎢ ⎥⎢ ⎥ − Θ +β Θ + γδγ⎣ ⎦ ⎣ ⎦

− Θ +α Θ +α δα δα⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Θ +α Θ +α δ δ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

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The minimum number of precision points usable for synthesis is limited by the number of loop closure equations (also called design equations).

Exact synthesis methods guide a point of interest exactly through the specified precision points but solutions exist only if the number of independent design equations is less than (or equal to) the number of design parameters. On the other hand, approximate synthesis methods minimize the structural error [17] to ensure least-squares satisfaction of the desired specification. This method is mainly used in over-determined geometric design problems where more precision points are defined than allowed for exact synthesis and therefore no exact solution exists.

The adjustable four-bar linkage can be synthesized by closed-form methods up to five precision points for path following problem [18]. Finding a solution for the four or more precision-point-synthesis problems involve nonlinear equations, hence we consider only the three precision point cases.

Figure 3. Kinematic synthesis with three precision points

The constraint equations used for 3PPS kinematic synthesis approach are derived from the loop-closure equation of adjustable four-bar mechanism (see Fig.3) as:

 

(3)

where and k2 and k3 are defined the stretch ratio of an adjustable four-bar mechanism. We can rewrite (3) in the matrix form, which yields

 (4)

where vector and are precision points of leg-wheel end-effector which they are determined by problem description (designer has a freedom in selecting the number and location of the precision points), and Z2, Z4, Z5, and Z6 are the position vectors of four-bar, M and N are position vectors of first link pivots on the chassis. The end point of four-bar is now guaranteed to pass through each of the selected precision points on the trajectory. Equation. (4) contains a total 8 scalar constraints obtained from the two scalar loop-closure per precision point in the path-following problem for each dyad with 16 scalar unknowns which they are Z1x, Z1y, Z2x, Z2y, Z3x, Z3y, Z4x, Z4y

. The stretch ratios k2, k3 are selected as the free choices due to the obvious physical interpretation of the extensibility of an adjustable link. By selecting , as free choices, and and a linear solution of the equations can be obtained from (4).

In this paper, we will consider an ideal step profile to have tread of 24cm and a rise of 16 cm (twice the wheel diameter) and. In general the desired kinematic motion curve consists of both the motion of the axle respect to the chassis and the motion of the body respect to the inertial frame. However, without loss of generality we will nominally assume the fixed chassis for the rest of this paper.

We set desired trajectory of wheel-axle to be the standard step profile as shown in Fig. 4, we set 220-150i as first precision point, 251.71+1.8i as second and put one more precision point at 270+10i for four-bar.

Figure 4. Leg-wheel configuration at precision points and initial

parameters To optimize the discrepancy between the desired and

actual curve, each curve should be divided into same length segments/points and calculate their distance. The objective

2 2

3 3

i i2 2 6 2 1

i i2 3 6 3 1

Z (k e 1) Z (e 1) P P

Z (k e 1) Z (e 1) P P

α β

α β

− + − = −

− + − = −2 2

3 3

i i4 5 2 1

i i4 5 3 1

Z (e 1) Z (e 1) P P

Z (e 1) Z (e 1) P P

γ β

γ β

− + − = −

− + − = −

6 3 5Z Z Z= +

2 2

3 3

2 2

3 3

i i4 2 1

i i5 3 1

i i2 2 12

i i6 3 13

Z P Pe 1 e 1 0 0Z P Pe 1 e 1 0 0Z P P0 0 k e 1 e 1Z P P0 0 k e 1 e 1

γ β

γ β

α β

α β

−⎡ ⎤− − ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−− −⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥−− −⎢ ⎥ ⎢ ⎥ ⎢ ⎥−− −⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦

1 2P ,P 3P

2 3 2 3, , ,α α γ γ 2 3 2 3, , , k ,kβ β

2 3,α α 2 3,β β 1 2P ,P 3P

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function is taken to be the structural error between the desired (Pxk) and actual path (Qxk) computed using arc length based correspondence points. The overall design problem can be expressed in the form of constrained problem as:

  (5)

subject to

 

(6)

As can be seen in the Fig.4, the end-effector can follow the desired curve, without needing to move the chassis back. The variation of stretch ratio and alpha are shown in Fig.5. The actual path passes through the edge of step which is one of the important locations to climb up step.

Figure.5 Stretch ratio for crank link adjustment (K) and the input

crank rotation ( )

C. Kinetostatic Synthesis The total required actuation torque (TActuate = TExt + TSpr) is derived by using the principle of virtual work for a system to be in equilibrium. The actuation torque depends on both the configuration and spring parameters.

  (7) where

  (8)

The expression for the torque due to external sources at the driving joint i.e. from Fx, Fy and Mz, and the weight of each link (TWeight) can be calculated from the expression for virtual work in the system as:

  (9)

where are derived from (2) and Ai, i = {1,2,…,8} are

defined as follows:

A5 = Kz2 cos(Θ2 +α)+C11 cos(Θ3 +β)z6 2

A6 = sin(Θ2 +α)z2 +C12 cos(Θ3 +β)z6 2

 

(10)

The linear torsional springs are used at each joint to reduce the required actuator torque (TActuate) on the driving joint. Let be the absolute initial configuration of each joint, is the absolute configuration of each joint at which each spring is assembled and corresponds to the relative displacement of the first link from its initial configuration to the current configuration. Then , the angular extension of each spring, can be written as:

  (11)

Therefore, the expression for the overall torque due to the spring torque TSpr can be expressed as:

 

(12)

The spring equations can also be rewritten in terms of the relative angular displacement of the each joint as a linear equation. The four design variables including {ai,bi,ci,di}, i=1,2 need to be selected for each actuator to match with desired static torque of linear actuator on the coupler link and joint actuator at the driving joint.

  (13)

where {ai,bi,ci,di} can be expressed as follows

( )cN

2 2xk xk yk ykDV k 1

Min (P Q ) (P Q )=

− + −∑

2 2

3 3

2 2

3 3

1i i4 2 1

i i5 3 1

i i2 2 12

i i6 3 13

2min 2 2max 3min 3 3max

min max min max

min max

Z P Pe 1 e 1 0 0Z P Pe 1 e 1 0 0Z P P0 0 k e 1 e 1Z P P0 0 k e 1 e 1k k k , k k k

,

−γ β

γ β

α β

α β

−⎡ ⎤− −⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥−− −⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥−− −⎢ ⎥⎢ ⎥ ⎢ ⎥−− −⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

≤ ≤ ≤ ≤α ≤ α ≤ α γ ≤ γ ≤ γβ ≤ β ≤ β

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.7

0.8

0.9

1

Time (sec)

Stre

tch

ratio

K, C

rank

Lin

k Ad

just

men

t

Stretch ratio and Joint Angular displacement

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

α (r

ad)

Stretch ratio K

α (rad)

α

δW = TActuateδη+ TExtδϑ + TSprδσ∑ = 0

TExtδϑ = Fxδx∑ + Fyδy∑ + Mzδφ∑ + FWeightδyG∑TActuateδη = TJointδα + FAdjustδK

[ ] [ ]

[ ]

[ ]( )[ ]( ) [ ]( )

x x 1 2 y y 3 4

z z 11 12

2 2 2 2 2Weight G

3 5 6 4 7 8

F x F A A , F y F A AK K

M M C CK

m g Kcos( ) z 2 sin( ) z 2F y

Km g A A m g A A

δα δα⎡ ⎤ ⎡ ⎤δ = δ =⎢ ⎥ ⎢ ⎥δ δ⎣ ⎦ ⎣ ⎦

δα⎡ ⎤δφ = ⎢ ⎥δ⎣ ⎦

⎛ ⎞Θ +α Θ +α + δα⎡ ⎤⎜ ⎟δ = ⎢ ⎥⎜ ⎟ δ+ ⎣ ⎦⎝ ⎠

∑ ∑

∑

∑

ijC

A1 = −z6 sin(Θ3 +β)C11 − Kz2 sin(Θ2 +α)( )A2 = z2 cos(Θ2 +α)− z6 sin(Θ3 +β)C12( )A3 = C11z6 cos(Θ3 +β)+ Kz2 cos(Θ2 +α)( )A4 = z2 sin(Θ2 +α)+C12z6 cos(Θ3 +β)( )A5 = Kz2 cos(Θ2 +α)+C11z6 cos(Θ3 +β)

A6 = sin(Θ2 +α)z2 +C12z6 cos(Θ3 +β)

A7 = C21 cos(Θ4 + γ )z4 2

A8 = C22 cos(Θ4 + γ )z4 2

A1 = −z6 sin(Θ3 +β)C11 − Kz2 sin(Θ2 +α)( )A2 = z2 cos(Θ2 +α)− z6 sin(Θ3 +β)C12( )A3 = C11z6 cos(Θ3 +β)+ Kz2 cos(Θ2 +α)( )A4 = z2 sin(Θ2 +α)+C12z6 cos(Θ3 +β)( )A5 = Kz2 cos(Θ2 +α)+C11z6 cos(Θ3 +β)

A6 = sin(Θ2 +α)z2 +C12z6 cos(Θ3 +β)

A7 = C21 cos(Θ4 + γ )z4 2

A8 = C22 cos(Θ4 + γ )z4 2

Θ i

iΩ, ,α β γ

iσ

( ) ( )( ) ( )

2 2 2 3 3 3 2 2

4 4 4 5 3 5 4 4

,

,

=Θ + −Ω = Θ + −Ω − Θ + −Ω

=Θ + −Ω = Θ + −Ω − Θ + −Ω

σ α σ β ασ γ σ β γ

( )[ ] ( )( )[ ] [ ]

2 2 2 9 11 12Spr

4 4 4 21 22 10 11 21 12 22

k 1 0 A C 1 CT

k C C A C C C C

⎛ ⎞Θ +α−Ω + − +⎡ ⎤⎣ ⎦⎜ ⎟= ⎜ ⎟Θ + γ −Ω + − −⎝ ⎠∑

( ) ( )( )( ) ( )( )

9 3 3 3 2 2

10 5 3 5 4 4

A k

A k

= Θ +β−Ω − Θ +α −Ω

= Θ +β−Ω − Θ + γ −Ω

Spr, 1 1 1 1

Spr,K 2 2 2 2

T a b c d

T a b c dα = α+ β+ γ +

= α+ β+ γ +

2863

 

(14)

and

   

(15)

The desired torque curves are assumed to be given as a function of for both actuators. We only have four design variables {ai,bi,ci,di} need to be calculated for each desired actuator torque curves in the static synthesis process of leg-wheel subsystem. We will employ the least-squares matching of a desired torque profile as the proposed objective in our work. No static precision points are used and this now allows us four variables as free choices. We minimize the discrepancy between desired and generated curve of the wheel-axle over the entire range of the motion. Therefore the optimization problem can be stated as:

  (16)

where

 

Figure.6 Actuator torque optimization for adjustable leg-wheel

subsystem Equation (16) includes eight constants ( 2 3k ,k , 4 5k , k ,

, ), and gives us a freedom for selection of spring constants and preloads. Spring preloads are selected as free variables and after the optimization the rest of

variables will be obtained. Figure. 6 shows the result of such an optimization where the final optimized torque profile, with spring assist, now matches the desired torque profile in the least squares sense. Now, for a given set of kinematic parameters, and a specified end-effector load profile, the torque required to equilibrate the load profile is approximately determined.

IV. ALTERNATE ADJUSTABLE LEG-WHEEL DESIGNS We examine the process by which we can make small

change to (3) to allow for creation of alternate adjustable design variants for (i) Ground (ii) Coupler or (iii) rocker adjustment.

A. Adjustable Ground Link Each prescribed precision point vector P1, P2 and P3 can be expressed as the summation of vectors Z1, Z4 and Z5 at the corresponding precision point. The kinematic synthesis (3) can now be rewritten as:

  (17)

where denotes the position displacement required when leg-wheel change from first to the second precision point and second to third, respectively.

Figure.7 Stretch Ratio of Ground Link

Figure.8 Stretch Ratio of Ground Link

B. Adjustable Coupler Link: In this case the magnitude of coupler link is adjustable and the parameters associated with this adjustment are the

( )( ) ( )

( )( ) ( )

( ) ( ) ( )( )( ) ( ) ( )( )

21 11

31 11 11 21

21 11 211 4

2 2 1 21 4 4 11 5

1 11 3 3 2 2

1 11 21 3 5 4 4

ka 1 C 1 0 0kb 0 C 1 0 C C

0 0 C C Cc kP C Qd k

P C 1

Q C C

− −⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ − −⎢ ⎥ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎢ ⎥⎢ ⎥ − −⎢ ⎥ ⎢ ⎥⎢ ⎥ θ −Ω θ −Ω⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

= − Θ −Ω − Θ −Ω

= − Θ −Ω − Θ −Ω

( )( )

( )

2 123

2 12 12 224

22 12 2225

2 22 4 4 22

a C 0 0k

b C 0 C Ck

0 C C Cck

P C Qd

−⎡ ⎤⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ − ⎢ ⎥⎢ ⎥⎢ ⎥ = ⎢ ⎥⎢ ⎥⎢ ⎥ − −⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦θ −Ω⎢ ⎥⎣ ⎦ ⎣ ⎦

( ) ( )( )( ) ( ) ( )( )

2 12 3 3 2 2

2 12 22 3 5 4 4

P C

Q C C

= Θ −Ω − Θ −Ω

= − Θ −Ω − Θ −Ω

,Kα

( ) ( )2 3 4 5

N 2 2des act des actjoint 1 Adjust 2

k ,k ,k ,k i 1T T T Tmin

=

− + −∑

act1 Ext,1 1 1 1 1

act2 Ext,2 2 2 2 2

T T a b c d

T T a b c d

= + α+ β+ γ +

= + α+ β+ γ +

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

-0.5

0

0.5

1

1.5

Time (sec)

Torq

ue(N

.m)

Balance Torque vs Desired Torque

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

-1

0

1

2

3

Time (sec)Exte

rnal

Tor

que

vs S

prin

g To

rque

s(N.

m) External Torques (including weight) vs Spring Torques

Balance TorqueDesired Torque

External TorquesSpring Torque

2 3,Ω Ω 4 5, ,Ω Ω

( ) ( ) ( )( ) ( ) ( )

2 2

3 3

i i2 1 4 5 2 1

i i3 1 4 5 3 1

1 Z Z e 1 Z e 1 P P

1 Z Z e 1 Z e 1 P P

γ β

γ β

λ − + − + − = −

λ − + − + − = −

2 3,λ λ

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time (sec)

Gro

und

link

stre

tch

ratio

λStretch ration of Ground link λ vs time

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

0

2

4

Time (sec)

Torq

ue(N

.m)

Balance Torque vs Desired Torque

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-20

-10

0

10

20

Time (sec)

Torq

ues

(N.m

)

External Torques (including weight) vs Spring Torques

Balance TorqueDesired Torque

External TorquesSpring Torque

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stretch ratios , . The kinematic synthesis (3) can be rewritten as

  (18)

Figure. 9 Stretch Ratio of Coupler Link

Figure. 10 External, Balance and Desired Torques for Adjustable

Coupler link

C. Adjustable Follower Link: Finally, in this case the kinematic synthesis (3) only require modification of stretch ratio associated with follower adjustment:

( ) ( )( ) ( )

2 2

3 3

i i4 2 5 2 1

i i4 3 5 3 1

Z e 1 Z e 1 P P

Z e 1 Z e 1 P P

γ β

γ β

η − + − = −

η − + − = −

  (19)

Figure. 11 Stretch Ratio of Follower Link

Figure. 12 External, Balance and Desired Torques for Adjustable

Follower Link The external, balance and desired torques along with the

stretch ratio for adjustable-ground-, coupler- and follower-link designs are shown in Figs. 7-12. In adjustable-ground-link design the actuators can be base-mounted but the payoff is significant variation of the stretch ratio (from 1 to -1 and vice versa). In contrast, the stretch ratio of the adjustable-follower-link (Fig.12) mechanism remains relatively monotonic and limited (from 1 to 0.7).

Figure 13. Crank, coupler, follower and ground link length variation

Figure. 13 shows the link length variation (defined as stretch ratio multiplied by nominal length) over the simulation time. It can be seen that, to perform the task, the adjustable-ground-link becomes negative for specific time intervals. This can be difficult to realize physically and hence not preferred. In addition, the variation of the adjustable-coupler-link can be quite significant which adversely affects the performance of the mechanism. Thus from this perspective crank length or follower length adjustments are better ways to physically realize the actuation torque reduction. A physical prototype system is being developed to aid the experimental verification of this process.

V. DISCUSSION In this paper, we proposed an adjustable articulated

2κ 3κ

( ) ( ) ( )( ) ( ) ( )

2 2 2

3 3 3

i i i2 3 5 2 2 1

i i i2 3 5 3 3 1

Z e 1 Z e 1 Z e 1 P P

Z e 1 Z e 1 Z e 1 P P

α β β

α β β

− + − + κ − = −

− + − + κ − = −

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.2

1.4

1.6

1.8

2

2.2

2.4

Time (sec)

Coup

ler lin

k stre

tch ra

tio

κ

Stretch ration of Coupler link κ vs time

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.5

0

0.5

1

Time (sec)

Torqu

e(N.m

)

Balance Torque vs Desired Torque

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

-5

0

5

10

Time (sec)

Torqu

es (N

.m)

External Torques (including weight) vs Spring Torques

Balance TorqueDesired Torque

External TorquesSpring Torque

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1.2

Time (sec)

Follo

wer li

nk st

retch

ratio

η

Stretch ration of Follower link η vs time

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

0

1

2

Time (sec)

Torqu

e(N.m

)

Balance Torque vs Desired Torque

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-4

-2

0

2

4

Time (sec)

Torqu

es (N

.m)

External Torques (including weight) vs Spring Torques

Balance TorqueDesired Torque

External TorquesSpring Torque

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-50

0

50

100

150

200

250

300

Time (sec)

Link L

ength

(mm)

Adjustable Links Extension

Coupler LinkCrank LinkFollower linkGround Link

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mechanism for leg-wheel subsystems and presented a general framework and methodology for their synthesis for planar tasks. The developed methodology allows the designer to specify a wheel–axle trajectory and prescribe forces along it to accurately match the kinematic and kinetostatic specifications at the precision points and in a least squares sense over the entire range. The kinetostatic design framework is a combination of kinematic design, structural equilibration design and design with spring assists, which permits the designer to arrive at optimal values for the mechanism parameters as well as the spring parameters simultaneously. The addition of springs substantially reduces the input torque requirements as demonstrated in figures.

Suitable selection of both kinematic parameters, such as the link lengths and initial configuration, as well as static parameters, such as spring constants and their preloads are critical. Finally, alternate designs of adjustable articulated leg-wheel subsystems were considered to modulate the motion and force interactions between the ground and the chassis and thereby enable vehicle systems to locomote in difficult environments and rough-terrain.

REFERENCES [1]. A. Bouloubasis, G. McKee, P.Tolson, “Novel concepts for a planetary

surface exploration rover”, Industrial Robots: An International Journal, Vol. 34(2), 2007, pp. 116–121.

[2]. M. Tarokh, G. J. McDermott, “Kinematics modeling and analysis of articulated rovers”, IEEE Transaction on Robotics, Vol. 21 (4), 2005, pp. 539-553.

[3]. S. Nakajima, “RT-Mover: a rough terrain mobile robot with a simple leg-wheel hybrid mechanism”, The International Journal of Robotics Research, Vol. 30 (13), 2011, pp. 1609-1626.

[4]. P. Morin, C. Samson, “Motion control of wheeled mobile robots”, Springer Handbook of Robotics, 2008, pp. 799-826.

[5]. H.W. Stone, “Mars pathfinder microrover: A low-cost, low-power spacecraft”, in: Proceedings of the 1996 AIAA Forum on Advanced Developments in Space Robotics, Madison, WI, 1996.

[6]. R. Volpe, J. Balaram, T. Ohm, R. Ivlev, “Rocky 7: A next generation Mars rover prototype”, Journal of Advanced Robotics. Vol. 11 (4), 1997, pp. 341–358.

[7]. T. Kubota, Y. Kuroda, Y. Kunii, I. Natakani, “Micro-planetary rover Micro5”, in: Proceedings of the Fifth International Symposium on Artificial Intelligence, Robotics and Automation in Space (ESA SP-440), Noordwijk, Netherlands, 1999, pp. 373–378.

[8]. R. Siegwart, P. Lamon, T. Estier, M. Lauria, and R. Piguet “Innovative Design for Wheeled Locomotion in Rough Terrain,” Robotics and Autonomous Systems, Vol.40(2), 2002, pp. 151–162.

[9]. D. Wettergreen, M. Bualat, D. Christian, K. Schwehr, et. al, “Operating Nomad During the Atacama Desert Trek,” Proceedings of Field and Service Robotics Conference, December 7–10, 1997, Canberra, Australia.

[10]. S.V. Sreenivasan, and K.J. Waldron, “Displacement Analysis of an Actively Articulated Wheeled Vehicle Configuration With Extensions to Motion Planning on Uneven Terrain,” ASME J. Mech. Des. 118(2), 1996 , pp. 312–320.

[11]. A. Halme, I. Leppänen, and S. Salmi, “Development of Workpartner- Robot—Design of Actuating and Motion Control System,” 2nd International Conference on Climbing and Walking Robots (CLAWAR 99), Portsmouth, England, Wiley Publishers, September 13–15,1999, pp. 657–666.

[12]. G. Endo, and S. Hirose, “Study on Roller-Walker Multi-Mode Steering Control and Self-Contained Locomotion” Proceedings of 2000 IEEE International Conference on Robotics and Automation, San Francisco, CA, April 24–28, 2000, pp. 2808–2814.

[13]. M. Hiller, and D. German, “Manoeuvrability of the Legged and Wheeled Vehicle ALDURO in Uneven Terrain with Consideration of Nonholonomic Constraints,” Proceeding off 2002 International Symposium on Mechatronics (ISOM 2002), March 21–22, 2002, Chemnitz, Germany.

[14]. S. K. Jun, G.D. White, V. Krovi, “Kinetostatic Design Considerations for an Articulated Leg-Wheel Locomotion Subsystem”, Transactions of the ASME, Vol. 128, 2006, pp. 112-121.

[15]. A. Alamdari, X. Zhou, and V. Krovi, “Kinematic Modeling, Analysis and Control of Highly Reconfigurable Articulated Wheeled Vehicles”, Proceedings of the ASME 2013 International Design Engineering Technical Conferences and Computers in Engineering Conference, ASME IDETC/CIE 2013, August 4-7, 2013, Portland, OR.

[16]. A. Alamdari, R. Herin, and V. Krovi, “Quantitative Kinematic Performance Comparison of Reconfigurable Leg-Wheeled Vehicles”, Proceedings of the 16th International Conference on Climbing and Walking Robots (CLAWAR 2013), July 14-17, 2013, University of Technology, Sydney, Australia.

[17]. Robert L. Norton, Design of Machinery: an Introduction to the synthesis and analysis of Mechanism and Machines, McGraw-Hill Inc., USA, 1992.

[18]. T. Chuenchom, S. Kota, “Synthesis of Programmable Mechanisms Using Adjustable Dyads”, Transactions of the ASME, Vol. 119, 1997.

[19]. J.F. McGovern, G.N. Sandor, “Kinematic synthesis of adjustable mechanisms (part 2: path generation)”, ASME Journal of Engineering for Industry, Vol. 95 (2), 1973, pp. 423–429.

[20]. C. Peng, R.S. Sodhi, “Optimal synthesis of adjustable mechanisms generating multi-phase approximate paths”, Mechanism and Machine Theory, Vol. 45, 2010.pp. 989–996.

[21]. P.V. Chanekar, A. Ghosal, “Optimal synthesis of adjustable planar four-bar crank-rocker type mechanisms for approximate multi-path generation”, Mechanism and Machine Theory 69, 2013, pp. 263–277.

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