Evaluation of Hip Kinematics Inﬂuence on the Performance of a

Quadrupedal Robot Leg

Navvab Kashiri, Arash Ajoudani, Darwin G. Caldwell and Nikos G. Tsagarakis

Department of Advanced Robotics, Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy

Keywords:

Leg Kinematics Design, Quadrupeds, Legged Robots, Force Manipulability/Polytope, Dynamic

Manipulability/Polytope.

Abstract:

As a major inspiration of biologically inspired systems, multi-legged robots have been developed due to their

superior stability feature resulting from their large support polygon. The leg design of a majority of such robots

is motivated by the skeleton of vertebrates such as dogs, or that of invertebrates such as spiders. Despite a

wide variety of multi-pedal robots on the basis of the two aforesaid leg designs, a thorough comparison of

the two underlying design principles remains to be done. This work addresses this problem and presents a

comparative study for the two mammal-like and spider-like designs by looking at the joint torque proﬁle, the

responsive motion of the legs, and the thrust force applied by the robot. To this end, a set of performance

indexes are deﬁned based on the gravity compensation torque, the dynamic manipulability polytope and the

force polytope, and evaluated in various leg conﬁgurations of the two designs.

1 INTRODUCTION

Robots have been widely employed in industrial

settings during the past four decades, offering a

range of functionality from manufacturing to food

processing. While a majority of such platforms are

manipulators mounted on ﬁxed/wheeled bases, there

are many scenarios that the robot needs to operate

in complex, unstructured, and dynamically changing

environments, where the human-/animal-like degrees

of mobility and agility would be essential (Caldwell

et al., 2014). Such a demand has created

new opportunities for the legged robots whose

functionality is not limited to naturally or artiﬁcially

smoothed terrains. Hence, an increasing worldwide

attention has been given to the development of pedal

robots within the past decade.

Even though the underlying anthropomorphic

structure of humanoid robots has illustrated the

potential of generating human-like mobility and

manipulation skills (Negrello et al., 2016; Bagheri

et al., 2015), the control of balance while performing

highly dynamic tasks or walking along rough terrains

is troublesome and still appears in a developing

stage. Multi-pedal robots (e.g. quadrupeds and

hexapods), on the other hand, present a higher level

of balancing performance due to a larger support

polygon compared to the bipedal ones.

The BigDog quadrupeds (Raibert et al., 2008)

developed by the Boston Dynamics, Inc., is a

renowned example of walking robots built for

outdoor applications, whose design is inspired by

the skeleton of mammals. The StarlETH (Hutter

et al., 2012), LittleDog (Shkolnik et al., 2010),

XDOg (Xie et al., 2014), MIT Cheetah (Seok

et al., 2013), IIT HyQ (Semini et al., 2011)

and Cheetah-cub (Spr

owitz et al., 2013) are other

examples of quadrupeds motivated by the mammalian

morphology. Alternative design of the robot legs is

inspired by the leg structure of insects and arachnids

such as spider (Ho et al., 2007; Gasparetto et al.,

2008), examples of which can be found in the design

of TITAN quadruped (Kato and Hirose, 2001), LAVA

(Zielinska and Heng, 2003), LAURON V (Roennau

et al., 2013), PUT Hexapod Robots (Belter et al.,

2015), MRWALLSPECT-III (Kang et al., 2003), and

MiniQuad I (Chen et al., 2008).

Despite the fast growing interest in the application

of the quadruped robots in various real-world

scenarios, the literature still fails to present a through

comparative study for the two aforementioned

design principles, especially regarding the dynamic

capabilities of the robot. Previous work in this area

mainly concerns kinematic considerations (Kar, 2003;

Chen et al., 2006; Zielinska, 2013), e.g. analysis of

the optimum inclination angle of the hip joint on the

Kashiri, N., Ajoudani, A., Caldwell, D. and Tsagarakis, N.

Evaluation of Hip Kinematics Inﬂuence on the Performance of a Quadrupedal Robot Leg.

DOI: 10.5220/0005986502050212

In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 1, pages 205-212

ISBN: 978-989-758-198-4

205

basis of kinematics manipulability (Roennau et al.,

2013). With that being said, this paper will attempt

to present a deeper analysis of the mammal-like

and spider-like legs for quadrupedal robots, although

it can be exploited for other pedal platforms in a

similar fashion. In this direction, a group of indexes

describing the statics and dynamics of the robot

is deﬁned and evaluated throughout the workspace

of the robot. The proposed indexes are based on

three criteria: the torque capacity computed from the

gravity compensation torque of joints representing the

static energy consumption of the system; a set of

measures derived from the dynamic manipulability of

legs describing the reactive capabilities of the robot;

and others extracted from the force manipulability

of legs analysing the capabilities of the robot to

avoid slippage. The study of the force/dynamic

manipulability is achieved through the deﬁnition of

the corresponding polytopes (Chiacchio et al., 1997),

to present the exact boundaries of the manipulator in

a given conﬁguration, subject to the actuator torque

bounds, in comparison to alternative less-accurate

representations such as ellipsoids (Yoshikawa, 1985).

The rest of this paper is organized as follows:

Section II describes the problem to be discussed

in this paper. The corresponding formulation is

presented in Section III, including the kinematics

and dynamics background, and the elaboration

of the three groups of criteria utilised for this

comparison study. Simulation analysis of the two

mammal-/spider- like leg design quadrupeds are

illustrated in Section IV, and the pertinent results

are discussed. Finally, Section V addresses the

conclusion and future works.

2 PROBLEM STATEMENT

A majority of quadrupedal robots make use of two

pre-dominant kinematics arrangements inspired by

the mammal or spider type legs, as noted in the

literature review. Concerning the ﬁrst three joints of

the leg related to the hip and the knee, they can be

described as follows

• mammal-like legs: A roll hip joint permitting the

abduction/Adduction motion is used for the ﬁrst

joint, in addition to two pitch joints for generating

ﬂexion/extension motion of hip and knee joints.

• spider-like legs: A yaw joint replicating

medial/lateral rotation is implemented on

the ﬁrst joint, followed by two pitch joints similar

to mammal-like arrangement.

The Schematics of quadrupeds with these two leg

(a) Roll-Pitch-Pitch leg (b) Yaw-Pitch-Pitch leg

Figure 1: Schematics of the two conventional kinematics

arrangement of quadrupedal robots. (The above images

are generated by means of the Matlab Robotics Toolbox

(Corke, 2011)).

arrangements are shown in Fig. 1. In this work, the

problem is set to realize the evolution of statics and

dynamics of these two types of quadrupedal robots,

so that the functionality of such robots for various

scenarios can be understood.

3 PROBLEM FORMULATION

3.1 Kinematics and Dynamics

Background

In this section the rigid body kinematics and

dynamics of non-redundant robotic linkages are

brieﬂy reported, see (Siciliano et al., 2009) for

details. For a given n-link body, the kinematic relation

expressing the end-effector position in the task space

based on joint coordinates can be expressed by a

vector function r ∈ ℜ

as follows

p = r(q), (1)

˙p = J(q) ˙q, (2)

where J ∈ ℜ

n×n

is the velocity Jacobian matrix,

deﬁned by J =

∂r(q)

∂q

, p ∈ ℜ

represents the

end-effector position in the task space, and the

superscript “T ” denotes the transpose operator. The

governing differential equations of this dynamical

system can be expressed by

M(q) ¨q + C(q, ˙q) ˙q + g(q) = τ

−J

ext

, (3)

where M(q) ∈ℜ

n×n

is the inertia matrices associated

with links; q = [q

, ..., q

] shows the vector of

generalized link positions; F

ext

∈ ℜ

is the vector of

forces applied by external objects/agents, τ

∈ ℜ

represents the vector of torques exerted by motors;

C(q, ˙q) ˙q ∈ ℜ

and g(q) ∈ ℜ

denote vectors of

Coriolis/centrifugal and gravitational torques of the

links, respectively.

Newton’s notation (over-dot) is used in this paper for

the presentation of derivatives with respect to time.

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206

3.2 Torque Capacity Measure: Gravity

Compensation Torque

The ﬁrst criterion on the leg design analysis is the

joints torques required in static conditions. Hence,

the robot is positioned in different conﬁgurations

within its workspace, and the gravity compensation

torques of joints are analysed. The maximum

torque of each joint required for holding the robot

in position, which is an essential measure for the

design, is therefore extracted. The evaluation is

carried out in two cases, when the robot stands on

a ﬂat surface and on an inclined surface. For the

sake of simplicity, legs are assumed be in mutually

symmetrical postures with respect to principal planes

described by XY Z-coordinate system, as shown in

Fig. 2(b). When the robot is positioned on a

horizontal plane, i.e. the surface slope γ = 0, the main

body mass is equally distributed between the four

legs. However, when the robot stand on an inclined

surface with a slope of γ, as shown in Fig. 2(a), the

ground reaction force applied on rear legs F

will

be larger than that on front legs F

for typical slope

angles 0 ≤γ ≤30

◦

. The rear leg gravity compensation

torque is thus used for the torque capacity evaluations.

3.3 Responsive Motion Index: Dynamic

Manipulability Polytope

For a given scenario in which the robot is subject to

external disturbances, the balance recovery requires

the placement of foot to right place using various

techniques (Pratt and Tedrake, 2006). The responsive

motions of the leg therefore plays a signiﬁcant role

in balancing of it, as well as in dynamic motions

of the robot. In order to evaluate the agility of

the robot in execution of such motions, the dynamic

manipulability concept is exploited. According to the

deﬁnition of the dynamic manipulability, proposed

and elaborated in (Yoshikawa, 1985), this index is

described on the basis of the relation between the

joint driving torque and the end-effector acceleration,

and it represents the robot capability of producing

(a) (b)

Figure 2: Schematics of the quadrupedal robot to be

discussed in this work, without demonstrating the ﬁrst joint

of the leg (hip yaw/roll).

arbitrary accelerations at the end-effector. To this

end, the dynamic manipulability polytope (DMP)

is exploited in this work to account for the exact

acceleration boundaries of the leg in task space, so

that the dynamic capability of legs in generating

instantaneous responsive motions can be precisely

illustrated. Since the main leg movement required for

the balance recovery of the robot should be carried out

within the plane in parallel to the ground, the dynamic

manipulability evolution in this plane will mostly be

focused, although the variation of this criterion in

vertical axis will also be evaluated.

To deﬁne the DMP, the joint torque polytope

deﬁning 2n bounds are expressed as follows

|τ

| ≤ τ

max

(4)

where τ

max

is the vector of maximum joint torques,

and |.| symbolizes the absolute value operator acting

on the components of the vector.

The DMP can therefore be derived by mapping

the vertices of the joint torque polytope. As it was

studied in (Yoshikawa, 1985; Chiacchio et al., 1997),

the mapping relation is extracted from (3) and the

second time derivative of (1) when assuming the robot

is standing still. It can therefore be expressed by

= M(q)J

−1

(q) ¨p + g(q). (5)

It should be noted that as the mapping relation

is linear and the joint torque polytope is convex,

the dynamic manipulability will hold the convex

property. By mapping the 2

vertices of the joint

torque polytope, the vertices required for constructing

the DMP is extracted, as shown in Fig. 3. At any given

conﬁguration, the DMP can therefore be computed

from the 2n equations as follows

|M(q)J

−1

(q) ¨p + g(q)| ≤ τ

max

. (6)

While the computation of DMP vertices for a system

with small number of degrees of freedom (DOFs)

can be done manually, for systems with large number

of DOFs, as well as for cases with high number

of iterations such as this work, when employing

Matlab

software, all mapped vertices can found

using the “lcon2vert” function, when the above-said

inequality (6) is expressed by A

¨p ≤ b

with



M(q)J

−1

(q)

−M(q)J

−1

(q)



, b



max

−g(q)

max

+ g(q)



. (7)

The DMP effective vertices can accordingly be

extracted and sorted by means of the “convhull”

function that also gives the area and the volume of

the polytope in 2D and 3D space, respectively. The

minimum distance from a given point and the 2D/3D,

to be used for further computations, can then be

Evaluation of Hip Kinematics Inﬂuence on the Performance of a Quadrupedal Robot Leg

207

(a) Joint torque polytope (b) DMP

Figure 3: A sample representation of the joint torque

polytope of a 3-DOF robot in a given conﬁguration, and

the corresponding dynamic manipulability polytope for the

aforementioned task space deﬁnition, with its coincidence

with the horizontal plane.

acquired by means of the “p poly dist” function. In

this work, the task space position vector is deﬁned by

p = [x, y, z]

, and three indexes are deﬁned as follows:

• The area of the DMP coincidence with the

horizontal plane xy; describing a criterion for

overall acceleration in this plane: A

• The ratio of the maximum to minimum

acceleration in the horizontal plane xy; subject

to spatial translation of the DMP due to the

gravitational torque; expressing the isotropy of

acceleration in this plane: α

• The ratio of the DMP volume to the area A

;

representing a measure for the acceleration along

the vertical axis z: a

3.4 Slippage Condition Index: Force

Polytope

While the prior index focused on the free motion of

the leg, this index evaluates the leg performance when

applying external force on the end-effector due to the

contact with ground. As the generation of thrust force

on the robot requires the leg end-effector to avoid

slipping on the ground, the third criterion studies the

slippage condition of the leg. The necessary and

sufﬁcient requirement for preventing an object from

slipping on a surface is to apply an adequately large

normal force that satisﬁes

≤ µ

, (8)

where F

and F

represent the tangential and normal

forces, respectively, and µ

is the static friction

coefﬁcient of the contact surface.

It is therefore suitable to design the robot leg in

such a way that the highest ratio of normal force to

tangential force on the end-effector can be achieved,

and the locomotion on a wider frictional range for

ground surfaces can be feasible. To achieve this, and

considering the higher accuracy obtained from force

polytope when compared to that from force ellipsoid,

the evolution of the normal force to tangential force

ratio is studied by means of force polytope. Based

on the actuator torque limits (4) presenting the joint

torque polytope, according to (Chiacchio et al., 1997),

the force polytope can be found when the joint

torque to the end-effector force mapping is applied

as follows

= J

ext

. (9)

The force polytope can therefore be computed

by implementing the same procedure as the DMP.

By applying the torque limit (4) on the above-said

torque-force mapping (9), the force polytope vertices

are extracted from A

ext

≤ b

with



(q)

−J

(q)



, b



max



, (10)

while b

is replaced by b

when the gravitational

translation of the polytope is taken into account.

As the robot body is assumed to be placed in

parallel with the ground, the tangential element F

is the external force in the horizontal plane, while

the normal element F

is the force along the vertical

axis. To demonstrate the variation in normal force to

tangential force ratio, the index β is deﬁned based on

the volume of the force polytope V

, the area of the

force polytope coincidence with the horizontal plane

, and the maximum tangential force F

as follows

β =

. (11)

4 SIMULATION ANALYSIS

In this section, the numerical simulations of two legs

based on the two kinematics arrangements are carried

out. For this comparison, the mass and length of

the leg links attached to the afore-stated joints are

assumed to be as follows: The bar linking the hip

roll/yaw joint to the hip pitch joint is 0.1

√

2 m long,

including 0.1 m offset and 0.1 m normal length, with

a mass of 2 Kg, the link connecting the hip pitch joint

to the knee joint has a length of 0.4 m and a mass of

4.5 Kg, and the last link attached to the knee joint is

0.4 m long with a mass of 3.5 Kg. The body mass is

considered to be 80 Kg. To facilitate the presentation

of results, the simulations are carried out in three

cases when the robot body is positioned at three

heights, and the Cartesian motion of the end-effector

is conﬁned with three half-circles as follows: I) at the

height of 0.3 m, with a half-circle of radius 0.5 m,

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208

Figure 4: The evolution of gravity compensation torque in xy-plane when the robot is set in different conﬁgurations within

the deﬁned workspace II, while standing on a horizontal plane γ = 0, for two kinematics arrangement: The RPP torques at the

ﬁrst row, and the YPP torques at the bottom.

Figure 5: The evolution of gravity compensation torque in

xy-plane when the robot is set in different conﬁgurations

within the deﬁned workspace I and III, while standing on

a horizontal plane γ = 0, for two kinematics arrangement:

The RPP torques on the left side, and the YPP torques on

the right side.

II) at the height of 0.5 m, with a half-circle of radius

0.4 m, III) at the height of 0.7 m, with a half-circle of

radius 0.3 m.

Fig. 4 presents the evolution of the gravity

compensation torque within the workspace deﬁned in

case II, for the both leg arrangements when standing

on a horizontal plane, γ = 0. It illustrates the torque

corresponding to individual joints, as well as the

−norm of the torque vector. It can be seen that,

when the robot operates on a horizontal plane, the

gravity compensation torques of the hip and knee

pitch joints vary within similar ranges, whereas the

ﬁrst joints of the two arrangements exhibit differently.

While the hip roll joint demands a considerable

amount of torque, the hip yaw joint needs a negligible

torque. The resulting torque norm of the YPP

arrangement is therefore larger than that of the RPP

arrangement. Such a difference can be also seen in

Table 1: Maximum gravity compensation torque for various

cases (values are in Nm).

||τ ||

Case I

RPP

γ = 0 74 75 65 99

γ = 30

◦

42 73 65 97

YPP

γ = 0 0 63 65 70

γ = 30

◦

36 54 65 70

Case II

RPP

γ = 0 54 65 65 91

γ = 30

◦

34 60 64 88

YPP

γ = 0 0 45 62 67

γ = 30

◦

28 49 65 80

Case III

RPP

γ = 0 40 50 55 75

γ = 30

◦

46 45 51 78

YPP

γ = 0 0 65 32 70

γ = 30

◦

28 56 56 68

the results corresponding to the workspaces deﬁned

in case I and III, reported in Fig. 5 (As for the cases

I and III, for the sake of brevity, only the norm

of the torque vector is reported here). However,

when the robot stands on an inclined surface with a

slope of γ = 30

◦

, see Figs. 6 and 7, the difference

between the maximum torque required for the ﬁrst

joints signiﬁcantly decrease. Nevertheless, the torque

threshold associated with the ﬁrst hip joint of the

YPP arrangement, as well as the torque vector norm,

remains lower than that of the RPP arrangement.

The maximum torque value corresponding to the

different cases are reported in Table 1, demonstrating

lower joint torques for the spider-like leg (YPP

arrangement), as compared to the mammal-like leg

(RPP arrangement). Based on these values, the torque

limit of joints are set to τ

max

= (75, 75, 65) Nm for the

further manipulability analysis.

The evolution of the DMP indexes within the

workspace cases I and III are illustrated in Figs. 8

and 9, respectively. It can be observed from

Fig. 8 that, when the robot body is relatively close

Evaluation of Hip Kinematics Inﬂuence on the Performance of a Quadrupedal Robot Leg

209

Figure 6: The evolution of gravity compensation torque in xy-plane when the robot is set in different conﬁgurations within

the deﬁned workspace II, while standing on an inclined plane γ = 30

◦

, for two kinematics arrangement: The RPP torques at

the ﬁrst row, and the YPP torques at the bottom.

Figure 7: The evolution of gravity compensation torque in

xy-plane when the robot is set in different conﬁgurations

within the deﬁned workspaces I and III, while standing on

an inclined plane γ = 30

◦

, for two kinematics arrangement:

The RPP torques on the left side, and the YPP torques on

the right side.

to the ground, the spider-like leg can possess a

higher level of dynamic manipulability isotropy than

the mammal-like leg; however, in terms of the

DMP magnitude in xy-plane, presented by A

, the

spider-like leg cannot show a better performance than

the mammal-like leg. On the other hand, when

the robot body is set to be in a higher distance

from the ground, i.e. the case III as depicted in

Fig. 9, the mammal-like leg exhibits higher dynamic

manipulability isotropy with lower magnitude A

as compared to the spider-like leg. Regarding

the dynamic manipulability along the vertical axis,

represented by a

, the mammal-like leg provides

larger end-effector accelerations than the other leg

arrangement in a majority of conﬁgurations.

The evaluation of the slippage condition index β

for the two leg kinematics arrangements is illustrated

in Fig. 10. It can be seen that when the robot

Figure 8: The evolution of the DMP indexes in xy-plane

when the robot is set in different conﬁgurations within the

deﬁned workspace I, for two kinematics arrangement: The

RPP torques on the left side, and the YPP torques on the

right side.

body is not positioned close to the ground (case

III), the mammal-like leg (RPP arrangement) allows

locomotion on more slippery surfaces, as compared to

the spider-like leg (YPP arrangement). However, the

spider-like leg can render motions on more slippery

surfaces when the robot is conﬁgured to hold its

body close to the ground (case I), although it is not

valid everywhere in the deﬁned workspace. When

the robot body is set in a position between the

two above-said cases (case II), the two RPP and

YPP arrangements exhibit similar performances when

the leg end-effector is not radially close to the hip

joint, while the mammal-like leg may possess better

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210

Figure 9: The evolution of the DMP indexes in xy-plane

when the robot is set in different conﬁgurations within the

deﬁned workspace III, for two kinematics arrangement: The

RPP torques on the left side, and the YPP torques on the

right side.

slippage avoidance capacity when the leg end-effector

radially approaches the ﬁrst joint.

5 CONCLUSIONS

This work presented a comparison study for

the leg kinematics arrangement of a quadrupedal

robot, on the basis of two conventional leg

conﬁgurations inspired by the skeleton of mammals

and spiders/insects. The statics and dynamics

performances of the two arrangements are evaluated

by deﬁning some criteria including the minimum

torque capacity of joints, the responsive/reactive

motion of legs, and the slippage avoidance condition.

To this end, a set of indexes are deﬁned based

on the gravity compensation torque of joints, the

dynamic manipulability polytope and the force

manipulability polytope. Eventually, the simulation

results describing the evolution of the performance

indexes within a set of workspaces are illustrated.

The results revealed the lower torque requirement

of the spider-like leg as compared to the mammal-like

leg, resulting from the perpendicularity of its ﬁrst

joint axis to the ground. It was also seen that

the mammal-like leg beneﬁts from higher dynamic

manipulability isotropy when the leg is nearly straight

and the robot body is close to its maximum distance

from the ground. In such a posture, the mammal-like

Figure 10: The evolution of the slippage avoidance index in

xy-plane when the robot is set in different conﬁgurations

within the deﬁned workspace I, II and III, presented at

the ﬁrst, second and third row, respectively; while the two

kinematics arrangement RPP and YPP are shown on the left

and right sides.

leg can also handle slippery surface better than the

spider-like leg. On the other hand, the spider-like leg

can possess larger dynamic manipulability isotropy,

and avoid slippage on more slippery surfaces, when

the robot body is positioned close to ground. The

mammal-like legs can therefore be selected when

straight leg dynamic motions are desired, while the

spider-like legs may be chosen if the execution of

high power tasks demanding large support polygon

and close-to-ground centre of mass is needed.

Future work of the authors may analyse the leg

performance when the passive/active impedance of

joints is taken into account (Laffranchi et al., 2014;

Kashiri et al., 2014; Spyrakos-Papastavridis et al.,

2015).

ACKNOWLEDGEMENTS

This project has received funding from the European

Union’s Horizon 2020 research and innovation

programme under grant agreement Centauro No

644839 (ICT-23-2014 Robotics).

Evaluation of Hip Kinematics Inﬂuence on the Performance of a Quadrupedal Robot Leg

211

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