Kinematic Analysis and Simulation of a Hybrid Biped Climbing Robot

Adri

an Peidr

o, Arturo Gil, Jos

e Mar

ıa Mar

ın, Yerai Berenguer and

Oscar Reinoso

Systems Engineering and Automation Department, Miguel Hern

andez University, 03202, Elche, Spain

Keywords:

Biped Robots, Climbing Robots, Hybrid Serial-parallel Robots, Kinematics, Redundant Robots, Simulation.

Abstract:

This paper presents a novel climbing robot that explores 3-D truss structures for maintenance and inspection

tasks. The robot is biped and has a hybrid serial-parallel architecture since each leg is composed of two

parallel mechanisms connected in series. First, the forward kinematic problem of the complete robot is solved,

obtaining the relative position and orientation between the feet in terms of the ten joint coordinates of the

robot. The inverse kinematics is more complex due to the redundancy of the robot. Hence, a simpliﬁed inverse

kinematic problem that assumes planar and symmetric movements is analyzed. Then, a tool to simulate the

kinematics of the robot is presented, and it is used to demonstrate that the robot can completely explore 3-D

structures, even when some movements are restricted to be planar and symmetric.

1 INTRODUCTION

Vertical structures such as buildings, bridges, silos,

or towers require periodic maintenance and inspec-

tion operations. For example, the glass facades of

skyscrapers must be cleaned, and the welded unions

in the metallic skeletons of the buildings must be ex-

amined. Tasks like these are very dangerous for hu-

man operators, who must work in environments of-

ten difﬁcult to access and are exposed to many risks

such as falling from height, contamination (e.g. in-

spections in nuclear or chemical facilities) or elec-

trocution (e.g. maintenance of power transmission

lines). To eliminate these risks, during the last two

decades many researchers have been investigating the

possibility of automating the execution of these tasks

using climbing robots. (Schmidt and Berns, 2013)

present an exhaustive analysis of the applications and

design criteria of climbing robots, as well as a com-

prehensive review of the main locomotion and adhe-

sion technologies.

Three-dimensional truss structures are present in

many vertical structures such as bridges, towers and

skeletons of buildings. These structures are typically

constituted by a network of beams connected at struc-

tural nodes, and a high degree of mobility is often

required to explore them. Climbing robots for 3-D

trusses can be classiﬁed into two main types (Tavakoli

et al., 2011): continuous-motion and step-by-step

robots. Continuous-motion robots are faster, use

wheels, and employ magnetism or friction to adhere

to the structure (Baghani et al., 2005; Tavakoli et al.,

2013). However, they usually have more difﬁcul-

ties to negotiate obstacles and their wheels may slip.

Step-by-step robots have two grippers connected by a

kinematic chain which has some degrees of freedom

(DOF). Their name reﬂects their locomotion method:

in each motion cycle, one gripper is ﬁxed to the struc-

ture, whereas the kinematic chain moves the other

gripper to the next attachment point of the structure,

where it will be ﬁxed. Then, the previously ﬁxed grip-

per is released and a new motion cycle begins. During

each motion cycle, these robots are equivalent to typ-

ical robot manipulators. Hence, they have a higher

mobility that facilitates the avoidance of obstacles,

but they are heavier, slower, and more complex.

The architecture of the kinematic chain of step-

by-step robots can be serial, parallel, or hybrid. Se-

rial architectures have larger workspaces than parallel

ones, but they are less rigid and have a limited load

capacity. The serial architectures have been the most

explored ones in step-by-step climbing robots, with

many different designs proposed by different authors.

For example, (Balaguer et al., 2000) present a 6-DOF

robot to explore 3-D metallic structures. Since the

robot is powered by a battery, the movements are opti-

mized to reduce the energy consumption and increase

its autonomy. Another 4-DOF serial climbing robot

is presented in (Tavakoli et al., 2011). Other authors

propose serial architectures inspired by inchworms,

with 5 and 8 DOF (Guan et al., 2011; Shvalb et al.,

2013). (Mampel et al., 2009) propose a similar mod-

Peidro A., Gil A., Marin J., Berenguer Y. and Reinoso O..

Kinematic Analysis and Simulation of a Hybrid Biped Climbing Robot.

DOI: 10.5220/0005515800240034

In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 24-34

ISBN: 978-989-758-123-6

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

ular robot whose number of DOF can be increased

connecting more modules in series. Finally, (Yoon

and Rus, 2007) present 3-DOF robots that can indi-

vidually explore 3-D trusses or can be combined with

other robots to form more complex kinematic chains

with higher maneuverability.

Parallel climbing robots are less common, but they

have also been studied. These architectures offer a

higher payload-to-weight ratio than serial robots, but

their workspace is more limited. (Aracil et al., 2006)

propose using a Gough-Stewart platform as the main

body of a robot for climbing truss structures, pipelines

and palm trees. The robot remains ﬁxed to the struc-

ture using grippers or embracing it with its annular

platforms.

Finally, hybrid climbing robots are composed of

some serially connected parallel mechanisms, and

they have the advantages of both architectures: high

maneuverability, rigidity and load capacity. A hy-

brid robot for climbing 3-D structures is proposed

by (Tavakoli et al., 2005), who combine a 3-RPR par-

allel robot with a rotation module connected in series.

Another hybrid robot is proposed in (Figliolini et al.,

2010). In this case, the robot is biped and each leg is

the serial combination of two 3-RPS parallel robots.

Hence, the complete robot has 12 DOF.

In this paper, we present a novel 10-DOF redun-

dant hybrid robot for climbing 3-D truss structures.

The robot is biped and its legs are connected to a hip

through revolute joints. Each leg is the serial combi-

nation of two parallel mechanisms that possess linear

hydraulic actuators, which provide a high load capac-

ity and stiffness. The design of the robot makes it

specially suitable to maneuver in 3-D truss structures

and perform transitions between planes with different

orientations. In this paper, we focus on the forward

and inverse kinematic problems of the robot, which

are necessary to plan trajectories in 3-D structures.

We also present a Java simulation tool that allows us

to verify the kinematic models obtained in this paper

and demonstrate the ability of the robot to explore 3-

D trusses.

This paper is organized as follows. The architec-

ture of the robot is described in detail in Section 2.

Next, the forward kinematic problem of the complete

robot when one foot remains ﬁxed is solved in Sec-

tion 3. In Section 4, a simpliﬁed yet useful version

of the inverse kinematic problem is solved. Then,

Section 5 presents a tool that simulates the forward

kinematics of the robot. This tool is used to demon-

strate the execution of some example trajectories by

the robot in a 3-D structure. Finally, the conclusions

and future work are exposed in Section 6.

2 DESCRIPTION OF THE ROBOT

Figure 1a shows a 3-D model of the biped climbing

robot. The robot has two identical legs (A and B)

connected to the hip through revolute joints driven by

motors (angles θ

and θ

). Each leg has three links:

a core link and two platforms. The lower platform is

the foot of the leg and carries the gripper that ﬁxes

the robot to the structure (the grippers are not consid-

ered in the kinematic analysis presented in this paper).

The upper platform is connected to the hip through

the aforementioned revolute joint. Each platform is

connected to the core link by means of two prismatic

actuators and a passive slider.

The mechanism composed of the core link, one

platform, and the two prismatic actuators that con-

nect these two elements, is a closed-loop linkage that

will be called hereafter “parallel module”. The paral-

lel modules are planar mechanisms that can be repre-

sented schematically as shown in Figure 1b. Hence,

each leg is the serial combination of the parallel mod-

ules 1 (which is connected to the foot) and 2 (which is

connected to the hip). The prismatic actuators of each

parallel module lie in opposite sides of the plane Π

which is one of the planes of symmetry of the core

link of the leg j (see the side view in Figure 1a). This

is indicated with dashed lines in Figure 2.

Figure 1a also shows some reference frames at-

tached to different parts of the robot. In this paper,

the X, Y , and Z axes of reference frames will be rep-

resented in red, green, and blue colors, respectively.

The frames H

and H

are ﬁxed to the hip of the robot,

whereas the frames A and B are respectively attached

to the feet of the legs A and B.

The robot has 10 DOF: the rotation angles θ

and

, and the four prismatic actuators of each leg. In

the next sections, the forward and inverse kinematic

problems of the robot will be analyzed. After that, we

will simulate the forward kinematics to demonstrate

its ability to explore 3-D structures.

3 FORWARD KINEMATICS

In this section, the forward kinematic problem (FKP)

of the robot is solved. The problem considered here

consists in calculating the position and orientation of

one foot with respect to the other foot when the joint

coordinates are known: the angles θ

and θ

and the

lengths (l

) of the linear actuators of the parallel

modules (i ∈

{

1,2

}

, j ∈

{

A,B

}

). First, the forward

kinematics of the parallel modules is analyzed.

KinematicAnalysisandSimulationofaHybridBipedClimbingRobot

hip

upper

platform

lower

platform

(foot)

motors

leg A

leg B

core

link

prismatic

actuators

passive

sliders

Side view

plane Π

core

link

(a)

base

(attached to the core

link of the leg j)

platform

(b)

Figure 1: (a) 3-D model of the climbing robot. (b) A schematic diagram of a parallel module.

3.1 FKP of the Parallel Modules

Figure 1b shows the i-th parallel module of the leg j

(i ∈

{

1,2

}

, j ∈

{

A,B

}

). A parallel module is a closed-

loop planar mechanism composed of a mobile plat-

form connected to a base through two prismatic ac-

tuators with lengths l

and r

. The platform is con-

strained to only translate vertically and rotate. The

forward kinematics consists in calculating the posi-

tion y

and the orientation ϕ

of the mobile platform

in terms of l

and r

. According to Figure 1b, the

relationship between (l

) and (y

,ϕ

) is:

(pcos ϕ

−b)

+ (y

+ p sin ϕ

)

= r

(1)

(pcos ϕ

−b)

+ (y

−p sin ϕ

)

= l

(2)

These equations can be combined to obtain an equiv-

alent system. Adding together Eqs. (1) and (2) yields

Eq. (3), whereas subtracting Eq. (2) from Eq. (1) re-

sults in Eq. (4):

4bpcos ϕ

= 2y

+ 2b

+ 2p

−l

−r

(3)

psinϕ

= r

−l

(4)

Solving cosϕ

from Eq. (3) gives:

cosϕ

+ 2b

+ 2p

−l

−r

4bp

(5)

Squaring Eq. (4):

16y

(1 −cos

) = (r

−l

)

(6)

Finally, substituting Eq. (5) into Eq. (6) yields a cubic

equation in ϒ

= y

+ k

= 0 (7)

where:

= 2b

+ 2p

−l

−r

(8)

(b + p)

−

+ r

(b − p)

−

+ r

(9)

= b

+ r

)

−r

)

/4 (10)

Equation (7) always has three roots, two of which may

be complex. For a given strictly positive root ϒ

Eq. (7), two solutions are obtained for y

= ±

For each of these two values of y

, cos ϕ

is calculated

from Eq. (5), whereas sin ϕ

is obtained from Eq. (4):

sinϕ

−l

(11)

Once cos ϕ

and sin ϕ

are known, ϕ

is unequivo-

cally determined in (−π,π]. If ϒ

= 0, then y

= 0 and

cosϕ

is calculated using Eq. (5). However, sin ϕ

cannot be calculated from Eq. (11) since y

= 0. In-

stead, sinϕ

is calculated as follows:

sinϕ

= ±

1 −cos

(12)

obtaining two solutions. It is shown in (Kong and

Gosselin, 2002), using Sturm’s Theorem, that Eq. (7)

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

cannot have more than two non-negative roots. Since

each non-negative root of Eq. (7) yields two different

pairs (y

,ϕ

), the FKP of each parallel module has

four solutions at most.

Note that swapping the values of r

and l

neither

affects Eq. (7) nor Eq. (5), but it changes the sign of

sinϕ

in Eq. (11). Hence, swapping r

and l

changes

the sign of ϕ

, leaving y

unchanged. This can be

observed in Figure 1b, where swapping r

and l

equivalent to rotating the ﬁgure π rad about the verti-

cal Y axis. This fact will be exploited in Section 4 to

analyze the inverse kinematics of the robot.

3.2 FKP of the Complete Robot

The forward kinematics of the complete robot con-

sists in calculating the position and orientation of

one foot with respect to the other foot when the ten

joint coordinates are known. The problem will be

solved using Homogeneous Transformation Matrices

(HTMs). An HTM has the following form (Bajd et al.,

2013):

m/n



m/n

1×3



(13)

where 0

1×3

= [0,0,0]. The matrix T

m/n

encodes the

position and orientation of a frame m with respect to

another frame n. Indeed, R

m/n

∈ R

3×3

is a rotation

matrix whose columns are the vectors of the frame m

expressed in the basis formed by the vectors of the

frame n, whereas t

m/n

∈ R

3×1

is the position of the

origin of the frame m in coordinates of the frame n.

core

link

hip

platform of the parallel

module 2

platform of the

parallel module 1

Figure 2: Kinematics of a generic leg j ∈

{

A,B

}

The forward kinematics of one leg can be easily

solved using HTMs. Figure 2 represents schemati-

cally a generic leg j ∈

{

A,B

}

. Each leg has two paral-

lel modules whose bases are attached to the core link.

The platform of the parallel module 1 is the foot of

the leg, whereas the platform of the parallel module

2 is connected to the hip of the robot by means of

a revolute joint. The variables (y

1 j

,ϕ

1 j

2 j

,ϕ

2 j

) are

obtained from (l

1 j

2 j

) as explained in Sec-

tion 3.1. All the reference frames of Figure 2 are con-

tained in the plane Π

, which is one of the planes of

symmetry of the core link of the leg j (see Figure 1a).

The transformation between the frame j (ﬁxed to the

foot) and the frame F

(ﬁxed to the core link) is:

/ j







cosϕ

1 j

sinϕ

1 j

0 y

1 j

sinϕ

1 j

−sinϕ

1 j

cosϕ

1 j

0 y

1 j

cosϕ

1 j

0 0 1 0

0 0 0 1







(14)

Similarly, the transformation between the frame G

(attached to the platform of the parallel module 2) and

the frame F

is:







cosϕ

2 j

−sinϕ

2 j

0 0

sinϕ

2 j

cosϕ

2 j

0 y

2 j

−h

0 0 1 0

0 0 0 1







(15)

where h is a geometric constant. Finally, a rotation θ

about the Y axis of the frame G

transforms it into the

frame H

, which is attached to the hip of the robot:







cosθ

0 sinθ

0 1 0 0

−sinθ

0 cosθ

0 0 0 1







(16)

The position and orientation of the frame H

with re-

spect to the frame j is obtained as follows:

/ j

= T

/ j

(17)

which completes the FKP of any generic leg j. Once

the forward kinematics of each leg is solved, it is

straightforward to calculate the position and orienta-

tion of the foot of one leg k ∈

{

A,B

}

{

}

with respect

to the foot of the other leg j:

k/ j

= T

/ j

k/H

(18)

where T

k/H





−1

and T

is the HTM that

encodes the position and orientation of the frame H

with respect to the frame H



I t

1×3



(19)

which is constant because both frames are attached to

the same rigid body (the hip). I is the 3 ×3 identity

matrix. Moreover, according to Figure 1a: t

[t,0,0]

= −t

, where t is the distance between

the parallel axes of the revolute actuators.

Note that, in theory, there are 4

= 256 different

solutions to the FKP of the complete robot. This is

because the kinematic chain between the feet has four

parallel modules connected in series and the FKP of

each module has four real solutions at most.

KinematicAnalysisandSimulationofaHybridBipedClimbingRobot

4 INVERSE KINEMATICS

The inverse kinematic problem (IKP) consists in cal-

culating the values of the joint coordinates necessary

to attain a desired relative position and orientation be-

tween the feet of the robot, and it is necessary for

planning trajectories. In this robot, ten joint coordi-

nates are used to place and orient one foot with respect

to the other foot, which makes it redundant. Hence,

the IKP is underconstrained and one should expect

inﬁnitely many solutions. This redundancy makes it

difﬁcult to solve the general IKP of this robot. For-

tunately, many important movements necessary to ex-

plore a 3-D structure (e.g., walking in one dimension,

changing between planes, etc) can be executed using

the conﬁguration analyzed in this section, which re-

duces the number of variables and simpliﬁes remark-

ably the IKP.

t/2

leg j

leg k

Figure 3: The Planar Symmetric Inverse Kinematic (PSIK)

problem.

The conﬁguration considered in this section is de-

picted in Figure 3, where the foot j is ﬁxed to the

structure and the foot k is mobile ( j,k ∈{A,B}, j 6= k).

It is assumed that the Z axes of the frames attached to

the feet are parallel and point in the same direction.

Hence, any variation in the length of the prismatic ac-

tuators of the parallel modules only produces planar

motions of the frame k in the XY plane of the frame j.

In this case, the position and orientation of the frame

k relative to the frame j can be calculated as follows:

k/ j

= T

/ j



I [t,0,0]

1×3







−1

(20)

where T

/ j

= T

/ j

. Moreover, it is assumed

that the joint coordinates of the parallel modules of

the two legs are related as follows:

= r

, r

= l

(i = 1, 2) (21)

This means that the joint coordinates of the parallel

module i of the legs k and j are swapped. According

to Section 3.1, this translates into:

= y

, ϕ

= −ϕ

(i = 1, 2) (22)

It can be graphically checked that Eq. (22) implies

that the legs k and j are symmetric with respect to the

line L, which is the axis of symmetry of the hip of the

robot. Substituting Eq. (22) into Eq. (20), the matrix

k/ j

can be written only in terms of the variables of

the leg j and has the following expression:

k/ j







−c(2ω) −s(2ω) 0 µ(1 −c(2ω))

s(2ω) −c(2ω) 0 µ ·s(2ω)

0 0 1 0

0 0 0 1







(23)

where s(x) = sin x, c(x) = cos x and:

µ =

t −2(h −y

1 j

−y

2 j

)sinϕ

2 j

2cos(ϕ

1 j

−ϕ

2 j

)

(24)

ω = ϕ

1 j

−ϕ

2 j

+ π/2 (25)

Thus, under the condition of planar and symmetric

motion, the position and orientation of the foot k rel-

ative to the foot j can be deﬁned by two parame-

ters (µ, ω), which are indicated in Figure 3. We de-

ﬁne the Planar Symmetric Inverse Kinematic (PSIK)

problem, which consists in calculating the joint co-

ordinates (l

1 j

2 j

) needed to achieve a desired

position and orientation (µ,ω). Since the joint coor-

dinates do not appear explicitly in Eqs. (24)-(25), the

kinematic equations of the parallel modules of the leg

j must be included:

(p cos ϕ

1 j

−b)

+ (y

1 j

+ p sin ϕ

1 j

)

= r

1 j

(26)

(p cos ϕ

1 j

−b)

+ (y

1 j

−p sin ϕ

1 j

)

= l

1 j

(27)

(p cos ϕ

2 j

−b)

+ (y

2 j

+ p sin ϕ

2 j

)

= r

2 j

(28)

(p cos ϕ

2 j

−b)

+ (y

2 j

−p sin ϕ

2 j

)

= l

2 j

(29)

Hence, the PSIK problem requires calculating

1 j

2 j

1 j

,ϕ

1 j

2 j

,ϕ

2 j

) from Eqs. (24)-(29).

Like the general inverse kinematic problem, the PSIK

problem is underconstrained since eight unknowns

must be obtained from six equations. However, the

PSIK problem involves less variables and simpler

equations. In the following section, we will show that

some postures necessary to negotiate obstacles in a

3-D structure can be analyzed solving the PSIK prob-

lem. Also, we will describe a method to choose ap-

propriate solutions to the PSIK problem assuming that

the lengths of the prismatic actuators of the parallel

modules have upper and lower limits.

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

leg A

leg B

Graphical window

Control window

Sliders for modifying the

joint coordinates

Button for changing

the fixed foot

Button for resetting

the simulator

Fixed foot

Output

window

Figure 4: Interface of the tool developed to simulate the forward kinematics of the robot.

5 SIMULATION

In this section, we will simulate the movements of

the complete robot in an example 3-D structure to

validate the kinematic analyses of Sections 3 and 4,

and demonstrate the ability of the robot to explore

the structure. More speciﬁcally, we will show how

the robot can walk on a beam, perform transitions

between different faces of the beams, and negotiate

structural nodes.

To demonstrate these movements, we have devel-

oped a Java simulation tool that can be downloaded

from http://arvc.umh.es/parola/climber.html (the lat-

est version of Java may be required). The simula-

tor implements the equations derived in Section 3 to

solve the forward kinematics. As shown in Figure 4,

the simulator has a graphical window that shows the

robot in the 3-D test structure. The tool also has a

window with a control panel where the user can mod-

ify the values of the ten joint coordinates, change the

foot that is attached to the structure, or reset the sim-

ulation. It is important to remark that the simulation

tool only implements the kinematic equations, with-

out considering the dynamics of the robot (gravity is

neglected) or the collisions between the robot and the

structure. These advanced topics will be analyzed in

the future.

Three reference frames are shown in the graphical

window of the simulator: the world frame W (which

is attached to one of the corners of the beam b

of the

structure) and the frames A and B (which are attached

to the feet of the legs). The ﬁxed foot is indicated

in orange color. When the user modiﬁes the value of

a joint coordinate, the forward kinematics is solved

and the position and orientation of the free foot with

respect to the frame W is calculated as follows:

k/W

= T

j/W

k/ j

(30)

where the matrix T

k/ j

is deﬁned in Section 3.2, j

denotes the ﬁxed leg, and k denotes the mobile leg

( j,k ∈

{

A,B

}

, j 6= k). As shown in Figure 4, the trans-

lation and rotation submatrices of T

A/W

and T

B/W

are

indicated to the user in an output window of the sim-

ulator. According to Section 3.2, there are 256 solu-

tions to the forward kinematics of the complete robot

since each parallel module can have up to four dif-

ferent solutions. However, it will be shown next that

only one solution is valid.

For the following simulations, we will assume

that b = p = 4 cm, and that the prismatic actuators

are constrained so that r

∈ [19,25] cm. Solving

the forward kinematics of a parallel module for these

ranges of the joint coordinates, and plotting the solu-

tion y

versus r

and l

, results in the four surfaces

shown in Figure 5. Each surface is associated with

one of the conﬁgurations labeled as follows: H

, X

−

, and X

−

. The solutions H

and X

are indicated

in Figure 5; the solutions H

−

and X

−

are their re-

spective mirror images with respect to the base of the

parallel module. According to the design of the robot

(see Section 2), the only valid solution is H

, since

the other solutions are impossible due to mechani-

cal interferences between different links of the legs.

Moreover, Figure 5 also provides a criterion for se-

lecting the valid solution: the solution H

always has

the highest y

coordinate.

Once the only valid solution to forward kinematics

has been characterized, we will simulate the execu-

tion of an example trajectory in the structure, which is

KinematicAnalysisandSimulationofaHybridBipedClimbingRobot

-30

-20

-10

(cm)

–

Figure 5: Solution surfaces of the FKP of a parallel module

for b = p = 4 cm. The surfaces H

and H

−

are almost

parallel to the surfaces X

and X

−

, respectively.

composed of the three beams b

, b

, and b

indicated

in Figure 4. At the beginning of the trajectory, the

robot lies on the face f

of the beam b

, and the objec-

tive is to move the robot to the face f

of the beam b

negotiating the structural node where the three beams

intersect. Next, we will show that such a trajectory

can be executed by a sequence of basic movements

that can be used to reach any other point of the struc-

ture. The values of the remaining geometric parame-

ters of the robot are: t = 15.6 cm, h = 16 cm. More-

over, the side of the square cross section of the beams

measures 12 cm, and the distance between the face f

of the beam b

and the origin of the frame W is 88 cm.

5.1 Phase 1: Walking Along a Beam

At the beginning of the trajectory (see Figure 6a), the

foot A is attached to the face f

of the beam b

, and the

frame A has the following position and orientation:

A/W





−40





cm, R

A/W





0 0 1

1 0 0

0 1 0





(31)

The number “6” in t

A/W

means that the frame A is

centered in the beam, whereas the number “5” is

a geometric constant of the feet of the robot. Ini-

tially, the joint coordinates have the following values:

= θ

= 0, r

= l

= 21 cm (i ∈

{

1,2

}

, j ∈

{

A,B

}

Starting from this conﬁguration, Table 1 describes a

simple sequence of movements that allows the robot

to reach the vertical beam b

. In each step of the given

sequence, we indicate only the joint coordinates that

change with respect to the previous step.

5.2 Phase 2: Concave Change of Plane

Once the beam b

has been reached, it can be climbed

to negotiate the structural node deﬁned by the inter-

section of the three beams. The next objective is to

perform a concave transition between the faces f

and

. Note that at the end of the previous phase (Fig-

ure 6f), the Z axes of the frames attached to the two

feet point in the same direction. Hence, the postures

needed to change between these faces can be obtained

solving the PSIK problem deﬁned in Section 4.

Figure 7a indicates the input parameters needed to

solve the PSIK problem: µ = 27.4 cm, ω = π/4 rad,

and j = B. Substituting these values and the geomet-

ric parameters of the robot into Eqs. (24)-(29) yields:

15.6 −2(16 −y

−y

)sinϕ

2cos(ϕ

−ϕ

)

= 27.4 (32)

−π/4 = ϕ

(33)

(4cosϕ

−4)

+ (y

+ 4 sin ϕ

)

= r

(34)

(4cosϕ

−4)

+ (y

−4 sin ϕ

)

= l

(35)

(4cosϕ

−4)

+ (y

+ 4 sin ϕ

)

= r

(36)

(4cosϕ

−4)

+ (y

−4 sin ϕ

)

= l

(37)

As discussed in Section 4, inﬁnitely many solutions

exist since there are eight variables to be solved from

six equations. Next, we describe a way of choosing

a proper solution to this underconstrained problem.

First, Eq. (33) is used to eliminate ϕ

from Eq. (32).

Then, ϕ

is solved from the resulting equation:

= sin

−1

13.7

√

2 −7.8

+ y

−16

(38)

This solution can be substituted into Eqs. (33)-(37)

to express the joint coordinates

{

}

terms of

{

}

, which can be chosen so that

∈ [19,25] (i = 1,2). Figure 8 represents the

curves of the (y

) plane in which each joint co-

ordinate equals 19 or 25; any point inside the shaded

region R enclosed by these curves is a valid solu-

tion to the PSIK problem. For example, the solu-

tion y

= y

= 22 cm yields: r

≈ 20.59536194,

≈ 23.40761347, r

≈ 23.65623783, and l

≈

20.34961301, all in cm (these accurate values are

valid only for the simulation; in a real implementa-

tion we will have to deal with the ﬁnite precision of

the sensors). This solution is used to perform a transi-

tion between the faces f

and f

(see Figure 7a). After

performing this transition, the foot A is attached to the

beam b

, and the sequence of movements described in

Table 2 is used to complete this phase.

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

12 cm

5 cm

6 cm

leg A

leg B

leg A

leg B

=19 cm

(a)

(b)

(c)

leg B

leg A

= π rad

=21 cm

leg B

leg A

leg B

leg A

=19 cm

The feet A and B are

released and fixed resp.

= π rad

leg B

leg A

(d)

(e)

(f)

Figure 6: Example trajectory where the robot moves along a beam of the structure.

Table 1: Sequence of movements in the ﬁrst phase of the simulated trajectory.

Step Joint coordinates Description of the movements in each step

1 r

= l

= 19 cm Retract the actuators connected to the foot B to lift it (Figure 6b).

2 θ

= π rad Rotate the robot about the leg A (Figure 6c).

3 r

= l

= 21 cm

Extend the actuators connected to the foot B until it touches the beam b

(Figure 6d).

4 r

= l

= 19 cm

Attach the foot B to the face f

Release and lift the foot A retracting the actuators connected to it (Figure 6e).

5 θ

= π rad Rotate the robot about the leg B (Figure 6f).

Table 2: Sequence of movements in the second phase of the simulated trajectory.

Step Joint coordinates Description of the movements in each step

= r

= 21 cm

= r

= 21 cm

= r

= 19 cm

Lift the foot B and place both legs perpendicular to the face f

, leaving

some distance between the foot B and the face f

(Figure 7b).

2 θ

= π/2 rad Rotate the leg B about its own axis (Figure 7c).

3 r

= l

= 21 cm

Extend the actuators connected to the foot B until it touches the face f

(Figure 7d).

4 r

= l

= 19 cm

Attach the foot B to the face f

Release and lift the foot A retracting the actuators connected to it (Figure 7e).

5 θ

= π rad Rotate the robot about the leg B (Figure 7f).

KinematicAnalysisandSimulationofaHybridBipedClimbingRobot

(e)

(b)

leg B

88 cm

5 cm

leg A

55.6 cm

leg A

leg B

leg A

leg B

π/2

leg A

leg B

leg A

leg B

=21 cm

=19 cm

leg A

leg B

π/2

(a)

(

)

Figure 7: A trajectory that includes a concave transition between different planes.

= 19

= 25

= 19

= 25

= 19

= 25

= 19

(cm)

Figure 8: Region of valid solutions to the PSIK problem.

5.3 Phase 3: Convex Change of Plane

At the end of phase 2, the Z axes of the frames at-

tached to the feet are parallel to the beam b

and

point in the same direction. Hence, the PSIK prob-

lem can be solved to determine the joint coordinates

that permit performing a convex transition from the

face f

to the face f

(the face f

is deﬁned in Fig-

ure 9). Substituting µ = 11 cm, ω = 3π/4 rad, and

j = B in Eqs. (24)-(29), and following the procedure

detailed in Section 5.2, we can obtain the region of

the (y

) plane where l

∈ [19,25] cm (i =

1,2). It can be checked that the solution adopted

in the previous section (y

= y

= 22 cm) is also

valid here, obtaining in this case: r

≈24.85374622,

≈ 19.20940403, r

≈ 21.99688208, and l

≈

22.00311791 (all in cm). For these values of the joint

coordinates, the robot can perform a transition be-

tween the faces f

and f

(see Figure 9a). After that,

the foot A can be attached to the face f

After attaching the foot A to the face f

, the se-

quence of movements described in Table 3 is exe-

cuted. After executing this sequence, solving exactly

the same PSIK problem as in Section 5.2 permits the

foot A of the robot to be attached to the face f

of the

beam b

, which completes the trajectory.

6 CONCLUSIONS

This paper has presented the kinematic analysis of

a novel biped climbing robot with a hybrid serial-

parallel architecture. The forward kinematic problem

was solved, obtaining the relative position and orien-

tation between the feet in terms of the joint coordi-

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

Table 3: Sequence of movements in the third phase of the simulated trajectory.

Step Joint coordinates Description of the movements in each step

= r

= 21 cm

= r

= 21 cm

= r

= 19 cm

Place both legs perpendicular to the face f

, leaving some distance between

the foot B and the face f

(Figure 9b).

2 θ

= 3π/2 rad Rotate the robot about the leg A (Figure 9c).

3 r

= l

= 21 cm

Extend the actuators connected to the foot B until it touches the face f

(Figure 9d).

4 r

= l

= 19 cm

Attach the foot B to the face f

Release and lift the foot A retracting the actuators connected to it (Figure 9e).

5 θ

= π rad Rotate the leg A about its own axis (Figure 9f).

(a)

(b) (c)

(d)

(e)

(f)

leg A

leg B

leg A

leg B

leg A

leg B

π/2

=21 cm

=19 cm

Figure 9: A trajectory that includes a convex transition between different planes.

nates. The inverse problem is more difﬁcult due to the

redundancy of the robot. Hence, a simpliﬁed inverse

problem was analyzed. It was shown that the simpli-

ﬁed problem is sufﬁcient to perform some important

trajectories which are necessary to explore 3-D struc-

tures. This was shown using a tool that simulates the

kinematics of the robot and demonstrates its ability to

explore 3-D trusses.

To exploit all the possibilities offered by the

proposed kinematic architecture, the general inverse

kinematic problem of the robot will be solved in the

future. Other problems that will need to be addressed

include the determination of the workspace (positions

and orientations that are attainable from a given at-

tachment point), the dynamic modeling of the robot,

and the planning of trajectories avoiding collisions.

Also, the performance of the robot will be studied

in more complex structures (with beams having ar-

bitrary orientation, not just orthogonal frames), and a

real prototype of the robot is currently being devel-

oped to test it in a real structure.

KinematicAnalysisandSimulationofaHybridBipedClimbingRobot

ACKNOWLEDGEMENTS

This work was supported by the Spanish Government

through the Ministerio de Educaci

on, Cultura y De-

porte under a FPU grant (Ref: FPU13/00413) and

through the Ministerio de Econom

ıa y Competitivi-

dad under Project DPI2013-41557-P.

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