# Adaptive Balancing of Robots and Mechatronic Systems

## Abstract

**:**

## 1. Introduction

**G**

_{p}. In case the load has variable weight (as is the case of oil pump-jack systems for example [5] then a more complex variation is possible (Figure 1b—solid curve line 1). A special situation is the one when the variation is known, and it is repeating during one cycle. In this case the adaptive solution could be a passive one (i.e., not actuated). Otherwise the balancing system should adapt in real time by using a local and supplementary actuation system and by aid of a controlling system and the required sensors and transducers [2].

## 2. Adaptive Balancing by Using Counterweights

**G**

_{1}and by the variable payload

**G**

_{p}has the expression:

**M**

_{g}(t) = −

**G**

_{1}OC

_{1}cos φ(t) −

**G**

_{p}(t) OP cos φ(t) =

**f**

_{1}(t) cos φ(t)

**f**

_{1}(t) = c

_{1}+ c

_{2}

**G**

_{p}(t)

_{1}= −

**G**

_{1}OC

_{1}= const. and c

_{2}= −OP = const.

**M**

_{b}=

**M**

_{b}(t) =

**f**

_{2}(t)

**f**

_{2}(t) ≅ −

**f**

_{1}(t) cos φ(t)

**G**

_{1}and for the weight of the constant part from the variation of payload

**G**

_{pc}(Figure 3) by a counterweight mounted fixed on the rocking arm ① at a proper distance on the opposite side then centre of mass C

_{1}according to origin point O (not represented in the following). In this case the constant c

_{1}from Relation (3) became:

_{1}= −

**G**

_{1}OC

_{1}−

**G**

_{pc}OP = const.

**G**

_{pv}by using also a supplementary counterweight then 2 possibilities could be taken into consideration: a variable weight of the additional counterweight or a movable counterweight with a fixed weight.

**M**

_{g}(t) = −

**G**

_{pv}(t) OP cos φ(t) = c

**f**

_{3}(t),

**G**

_{pv}(t) =

**G**

_{p}(t) −

**G**

_{pc}

**M**

_{b}of counterweight ② has the expression:

**M**

_{b}(t) =

**G**

_{2}OB(t) cos φ(t),

**G**

_{2}could be count as added the part of the weight of the connecting bar ③ concentrated in point B because is fixed one (Figure 3a).

^{2}(t) = OA

^{2}+ AB

^{2}− 2 OA AB cos ϕ(t)

**M**

_{u}is given by relation:

**M**

_{u}=

**M**

_{b}+

**M**

_{g},

- -
- the position of points A and B;
- -
- the length of bars BC and AB.

- -
- coordinates of points A(x
_{1A}, y_{1A}) and C(X_{C}, Y_{C}); - -
- lengths of connecting bars AB and BC;
- -
- position AC
_{2}of the counterweight on the bar ② and the mass of the counterweight m_{2}.

## 3. Adaptive Balancing by Using Springs

**G**

_{p}is done then joint A and joint B respectively, are fixed to the arm and to the ground respectively.

_{1}axis and also the point of action of payload in same point A. In this case the equilibrium equation of rocking arm ① is given by equation:

**F**

_{s}OA sin(θ − φ

_{1}) −

**M**

_{g}

_{1}= 0

**M**

_{g}

_{1}= (m

_{1}OC

_{1}+ m

_{p}OA) g cos φ

_{1}

**F**

_{s}=

**F**

_{s}

_{0}+ k (l

_{s}− l

_{s}

_{0}),

- -
- force of spring
**F**_{s}_{0}corresponding to the length of spring l_{s}_{0}, - -
- coordinates: x
_{1A}, X_{B}and Y_{B}.

**G**

_{p}′ =

**G**

_{p}+ Δ

**G**

_{p}or m

_{p}′ = m

_{p}+ Δm

_{p}

_{B}′ = Y

_{B}+ ΔY

_{B}

**F**

_{s}′=

**F**

_{s}

_{0}+ k (l

_{s}′ − l

_{s}

_{0}) =

**F**

_{s}+ k Δl

_{s},

**F**

_{s}′ OA sin(θ′ − φ

_{1}) −

**M**

_{g}

_{1}− Δ

**M**

_{g}= 0

**M**

_{g}= Δm

_{p}g OA cos φ

_{1}= Δ

**G**

_{p}OA cos φ

_{1}

_{B}′ = Y

_{B}′(m

_{p}(t))

_{B}= ΔY

_{B}(Δm

_{p}(t))

## 4. Example

**G**

_{pv}

_{,max}= 4 N (Figure 3) and is acting at distance OP = 2 m while the work space of balanced arm ① is symmetric with respect to the horizontal axis: φ ∈ [−π/2, π/2]. Suppose that the counterweight ② has the weight

**G**

_{2}= 3 N and the connecting road ③ has the length AB = 2 m and is articulated on vertical direction at distance OA = 1 m.

_{2}(x) plotted in red color in graph from Figure 9).

_{1}(x) in Figure 9—show the variation of unbalancing moment in case o linear variation of static load [30] which has the maximum value about double than in case of parabolic variation (about 1.4 Nm at position φ = −0.5 rad).

**E**—energy consumed by actuating system of the balanced robot;_{b}**E**—supplementary energy consumed by an additional actuating system in order to obtain active balancing of robot;_{a}**E**—energy consumed by actuating system of the unbalanced robot._{u}

**E**is zero and in present example by comparing the other energies without taking into the consideration the frictions the following efficaciousness coefficient are obtained:

_{a}_{linear}= 0.632031

_{parabolic}= 0.973446

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Industrial robot static balanced by counterweight and spring: (

**a**) kinematic schema; (

**b**) picture of industrial robot.

**Figure 4.**Movable counterweight in order to compensate variable payload: (

**a**) translational counterweight; (

**b**) rocking counterweight.

**Figure 6.**Adjusting mechanisms with revolute pairs: (

**a**) Controlled relocation of joint A of spring; (

**b**) Controlled relocation of fixed joint B of spring.

**Figure 7.**Adjusting mechanisms with revolute and prismatic pairs: (

**a**) Controlled relocation of fixed joint B of spring; (

**b**) Controlled relocation of joint A of spring.

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Ciupitu, L. Adaptive Balancing of Robots and Mechatronic Systems. *Robotics* **2018**, *7*, 68.
https://doi.org/10.3390/robotics7040068

**AMA Style**

Ciupitu L. Adaptive Balancing of Robots and Mechatronic Systems. *Robotics*. 2018; 7(4):68.
https://doi.org/10.3390/robotics7040068

**Chicago/Turabian Style**

Ciupitu, Liviu. 2018. "Adaptive Balancing of Robots and Mechatronic Systems" *Robotics* 7, no. 4: 68.
https://doi.org/10.3390/robotics7040068