The energy consumption of walking robots has been considered on the base of solution of variational problem. It is fixed that the gait is affected on the energy efficiency and must vary with speed

Eugene S. Briskin

Yaroslav V. Kalinin

English

Vibroengineering PROCEDIA

 346 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
Effect of gait on the energy consumption of walking 
robots 
Eugene S. Briskin1, Yaroslav V. Kalinin2 
Volgograd State Technical University, Volgograd, Russia 
2Corresponding author 
E-mail: 1dtm@vstu.ru, 2jkv83@mail.ru 
(Received 7 September 2016; accepted 12 September 2016) 
Abstract. The energy consumption of walking robots has been considered on the base of solution 
of variational problem. It is fixed that the gait is affected on the energy efficiency and must vary 
with speed. 
Keywords: walking robot, walking mechanism, energy efficiency, energy consumption, heat 
losses, variational problem, gait of walking robot. 
1. Introduction 
One of the problems of walking robots is their low energy efficiency. It is known [1] that 
energy consumptions grow in proportion to the cube of speed when walking robot moves with 
constant velocity of its body. This is due with the periodic acceleration and deceleration of 
unbalanced walking mechanisms in the transfer phase from one position to another. The use of 
energy recovery allowing to accumulate energy in the braking phase of the walking mechanism 
in the transfer and then to give it in the phase of accelerating, allows to increase the energy 
efficiency [2].  
It is known methods to improve energy efficiency at the expense of motion comfort as bipedal 
[3] and multilegged robots [4]. For multilegged robots with dual walking mechanisms resulting 
equation gives you the opportunity to choose such law of motion, which ensures low energy 
consumption [5]. This equation for walking movers operating from a common motor has a form: 
ሶܶ + ܸܳ = ܳ௔ ௔ܸ, (1)
where ܶ is kinetic energy of robots body and its movers; ܳ, ܸ, ܳ௔, ௔ܸ - according resistance force 
and the velocity of the body translational motion in the current time and their average values in 
the one cycle.  
In the general case the Eq. (1) is nonlinear differential equation of the second order with respect 
to the coordinates. 
2. Statement of the problem 
As we know there is a fairly complete classification of possible gaits of multilegged walking 
robots during their translational motion with rectilinear and uniform motion of body center of 
mass. The ratio mode ߛ is the important element of classification: 
ߛ = ߬ − ݐ௜߬ , (2)
where ݐ௜ is the time spent of walking mechanism in the phase of interacting with supported surface, 
߬ is total cycle time. 
The another important feature is the time ௝߬ of the start transfer of ݆ mover to a new position. 
The countdown is since the beginning of shifting phase of one of the walking mechanism.  
This mechanism is considered as baseline ௝߬ = 0.  
Given that of the beginning and end of the transfer, ݆ mover it absolute speed: 
EFFECT OF GAIT ON THE ENERGY CONSUMPTION OF WALKING ROBOTS.  
EUGENE S. BRISKIN, YAROSLAV V. KALININ 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 347 
ݔሶ௝ = 0, (3)
the motion of the body is performed without sliding of its pads on the ground. 
The possible dependence ݔሶ௝ = ݔሶ௝(ݐ) are shown in the Fig. 1. 
 
Fig. 1. Velocities of movers for example gait (߬ = 0.5 second): 
1 – The baseline walking mechanism ௝߬ = 0, 2 – ݆ walking mechanism in case ௝߬ > 0,  
3 – ݆ walking mechanism in case ௝߬ < 0. 
The simplest analytical dependence of the absolute speed ݔሶ௝ with time and satisfying the set 
conditions in Eq. (3) has the form: 
ݔሶ௝ = ቐ−
6ܵ
ݐ௜ଷ
൫ݐ − ௝߬൯ଶ +
6ܵ
ݐ௜ଶ
൫ݐ − ௝߬൯, ௝߬ < ݐ < ௝߬ + ݐ௜,
0, ݐ < ௝߬, ݐ > ௝߬ + ݐ௜,
(4)
ݔሶ௝ = ቐ−
6ܵ
ݐ௜ଷ
൫ݐ − ௝߬൯ଶ +
6ܵ
ݐ௜ଶ
൫ݐ − ௝߬൯, ߬ + ௝߬ < ݐ,  ݐ < ௝߬ + ݐ௜,
0, ௝߬ + ݐ௜ < ݐ < ௝߬ + ߬,
(5)
where ܵ is the distance at which the walking mechanism pad is transferred over time ߬ − ݐ௜.  
The main goal of submitted paper is to define such a low of rectilinear motion of the robot 
center of mass during its forward motion which provides the minimum of total heat losses all 
engines that control motion of all movers at various gaits and speeds. 
3. Design scheme and mathematical model of walking robot 
The translational motion of walking robot body of mass ܯ and ܰ its movers of mass ݉ each 
are studied. These motions are described by absolute coordinates ݔ௖ , ݔ௝ (Fig. 2) at that ௝ܽ  – 
coordinate of ݆  mover installation, ݖ௝  – coordinate ݆  mover with respect to the point of its 
installation. 
 
Fig. 2. Design scheme of walking robot 
The resistance force ܳ and forces of interactions between the body of walking robot and their 
movers ܨ௝ are generated by drives of the course motion. The interaction of ݆ mover with supported 
EFFECT OF GAIT ON THE ENERGY CONSUMPTION OF WALKING ROBOTS.  
EUGENE S. BRISKIN, YAROSLAV V. KALININ 
348 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
surface is taken into account by tangent forces ௝ܴ . The supported surface is modeled by 
homogeneous elastic medium and the body of the robot is modeled by absolutely solid. For this 
reason, all reactions ௝ܴ for movers interacted with a ground are equal each other: 
௝ܴ = ܴ. (6)
Then the differential equations, described the motion of considered mechanical system have a 
form: 
ܯݔሷ௖ = ෍ ܨ௝ − ܳ, ݆ = 1,2, … , ܰ, 
0 = ܴ − ܨ௝, ݆ = 1,2, … , ݇, 
݉ݔሷ௝ = −ܨ௝, ݆ = ݇ + 1, ݇ + 2, … , ܰ, 
(7)
where ܰ is the total number of movers, ݇ is the number of movers interacted with the support 
surface in a given moment of time and dependent on the gait. It follows from Eq. (7) that the force, 
generated by actuator is determined from second equation for movers interactive with the support 
surface and for movers which are in phase of transfer the force is determined from third. The 
constant force ܴ is determined from the first Eq. (7): 
ܴ = ܯݔሷ௖ + ܳ + ∑ ݉ݔሷ௝
ே௝ୀ௞ାଵ
݇ . (8)
The feature of the Eq. (8) is the fact that in the process of motion a number of movers ݇ interact 
with the supporting surface may change. Hence considered mechanical system is system of 
variable structure.  
4. Method of solution 
In accordance with the task the method of solution is based on the requirement of the minimum 
of integral. 
ܫ∗ = න ෍ ܨ௝ଶ
ே
௝ୀଵ
݀ݐ
ఛ
଴
. (9)
It is taken into account the isoperimetric condition: 
න ݔሶ௖݀ݐ
ఛ
଴
= ܵ, (10)
where ܵ is the distance, traversed by the body of robot during the time of interacting of walking 
mechanism with a supported surface. 
Then we can obtain a new functional ܫ by combining Eq. (9) and Eq. (10): 
ܫ = න ቌ෍ ܨ௝ଶ + ߤݔሶ௖
ே
௝ୀଵ
ቍ ݀ݐ = න Φ݀ݐ
ఛ
଴
ఛ
଴
. (11)
Φ = ෍ ቆܯݔሷ௖ + ܳ + ∑ ݉ݔሷ௝
ே௝ୀ௞ାଵ
݇ ቇ
ଶ௞
௝ୀଵ
+ ෍ ൫݉ݔሷ௝൯ଶ
ே
௝ୀ௞ାଵ
+ ߤݔሶ௖, (12)
EFFECT OF GAIT ON THE ENERGY CONSUMPTION OF WALKING ROBOTS.  
EUGENE S. BRISKIN, YAROSLAV V. KALININ 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 349 
where ߤ is undetermined Lagrange multiplier. 
Since ݔሷ௝  are defined as time function Eq. (4) and Eq. (5) and parameters ܵ , ݐ௜ , ௝߬ , the 
Euler-Poisson equations [6], providing the minimum of functional has a form: 
݀
݀ݐ ൬
∂Φ
∂ݔሷ௖൰ −
∂Φ
∂ݔሶ௖ = ܥ, (13)
or in expended form: 
2 ܯ
ଶ
݇ ݔഺ௖ + 2
ܯ
݇ ෍ ݉ݔഺ௝
ே
௝ୀ௞ାଵ
= ߤ + ܥ, (14)
where ܥ is arbitrary constant. 
The Eq. (14) must satisfy the initial and final conditions: 
ݔ௖(0) = 0, ݔ௖(τ) = ܵ, ݔሶ௖(0) = ݔሶ௖(τ) = ଴ܸ, (15)
which allow to determine three arbitrary constants of differential equation and the amount ߤ + ܥ, 
଴ܸ – initial and final velocity of robot’s body in each cycle. 
One more peculiarity of Eq. (14) is dependence of the amount ݇ of movers interacting with a 
support surface from the gait. 
5. Examples of the influence of gait on energy efficiency 
The real walking robot “Ortonog” [7] (Fig. 3) have been considered as example of walking 
robot. 
 
Fig. 3. Walking robot “Ortonog” 
For this robot four mechanisms of walking always are interacted with the support surface. It is 
considered a such gait when the interacting with a support surface is carried out in pairs of two 
mechanisms of walking with ratio mode ߛ  > 0.5. It is presented on chart in Fig. 4 the 
dimensionless parameter ߝ: 
ߝ = ܫ݉ଶ݃ଶ߬ , (16)
where ߝ is parameter proportional energy losses, depending on ratio mode and average speed of 
robot, ݃ is acceleration of gravity. 
The results show us that in order to minimize the energy consumption it is necessary to change 
the gait. This is confirmed by theory of synchronization [8], behavior of animals and investigation 
of multilegged walking machines and robots motion [9]. 
EFFECT OF GAIT ON THE ENERGY CONSUMPTION OF WALKING ROBOTS.  
EUGENE S. BRISKIN, YAROSLAV V. KALININ 
350 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
 
Fig. 4. Dependencies of the rate of energy loss of walking robot’s speed and gait  
1 – ଴ܸ is 0.5 m/s, 2 – ଴ܸ is 0.33 m/s, 3 – ଴ܸ is 0.25 m/s 
6. Conclusion 
The gait of walking robot significant effects on the energy consumption. For each speed you 
should choose energy optimal gait, but the most effective use of such actuator together with control 
system which could independently change the gait. 
Acknowledgements 
The work was supported by the RFBR (Grants Nos. 14-01-00655-A, 14-08-01002-A) and by 
Grant of Ministry of Education and Science of Russian Federation No. 862. 
References 
[1] Ohotsimsky D. Е., Platonov A. K., Kiril’chenko А. А., Lapshin V. V. Schagayuschie Maschiny 
(Walking Machines). Preprint IPM of M.V. Keldysch AN USSR, Moscow, 1989, (in Russian). 
[2] Lapshin V. V. Modelnye Ocenki Energozatrat Schagayuschego Apparata (Model Estimates of Energy 
Walking Machine). Izvestia RAS, МТТ (News of RAS. Mechanics of Solid Body), Vol. 1, 1993, 
p. 65-74, (in Russian). 
[3] Beletskiy V. V. Dvunogaya Hod’ba: Model’nye Zadachi Dinamiki i Upravleniya (Biped Walking: 
Modeling Problems of Dynamics and Control). Science, Moscow, 1984, (in Russian). 
[4] Briskin Е. S., Kalinin Ya V. On energetically efficient algorithms of walking machines with cyclic 
drives. Journal of Computer and Systems Sciences International, Vol. 50, Issue 2, 2011, p. 348-354. 
[5] Briskin Е. S., Chernyschev V. V., Kalinin Ya. V., Маloletov А. V. On the energy efficiency of 
cyclic mechanisms. Mechanic of Solids, Vol. 49, Issue 1, 2014, p. 11-17. 
[6] Halfman R. L. Dynamics. Addison-Wesley Publishing Company, Inc. Reading, Massachusetts, Palo 
Alto, London, 1962. 
[7] Briskin Е. S., Maloletov А. V., Sharonov N. G., Fomenko S. S., Kalinin Ya. V., Leonard А. V. 
Development of rotary type movers discretely interacting with supporting surface and problems of 
control their movement. Proceedings of the 21st CISM-IFToMM Symposium on Robot Design, 
Dynamics and Control, Udine, Italy, 2016, p. 351-359. 
[8] Blekhman I. I. Synchronization in Science and Technology. ASME Press, New York, 1981. 
[9] Briskin Е. S. Ob Obschei Dinamike i Povorote Shagayuschih Mashin (On General Dynamics and 
Turns of Walking Machines). Problemy Mashinostroeniya i Dinamiki Mashin (Problems of 
Engineering and Reliability of Machines), Vol. 6, 1997, p. 33-39, (in Russian). 


Effect of gait on the energy consumption of walking robots

https://www.jvejournals.com/article/17693/pdf

Effect of gait on the energy consumption of walking robots

Abstract

Similar works

Full text

Available Versions

Vibroengineering PROCEDIA