CN109782759A - A kind of Approximate Decoupling of wheeled mobile robot, quick Trajectory Tracking Control method - Google Patents
A kind of Approximate Decoupling of wheeled mobile robot, quick Trajectory Tracking Control method Download PDFInfo
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Abstract
A kind of Approximate Decoupling of wheeled mobile robot, quick Trajectory Tracking Control method, include the following steps: 1) to establish error model and the error differential equation according to the kinematics model of wheeled mobile robot and desired trajectory model；2) new control law is used, the forward speed of wheeled mobile robot and the expression formula of steering angular velocity are provided；3) parameter of expression formula is set to require to meet decoupling and quickly adjust.The contrail tracker proposed has many advantages, such as that simple control structure, controlled device Approximate Decoupling, adjustment parameter are few, fast convergence rate.
Description
Technical field
The present invention relates to a kind of Approximate Decouplings of wheeled mobile robot, quick Trajectory Tracking Control method, especially relate to
A kind of and Trajectory Tracking Control method of simple and effective, control decoupling the wheeled mobile robot of control structure.
Background technique
Trajectory Tracking Control is one of important content in Control of Wheeled Mobile Robots.The method of Trajectory Tracking Control has
Very much, such as: Lyapunov direct method control, Sliding mode variable structure control, Backstepping control etc..Although the above method can reach
To tracing control requirement, but controller destructing is complicated, and lacks between controller parameter and controlled volume and explicitly adjust relationship, i.e.,
Some parameter is adjusted, multiple controlled volume performances may be influenced simultaneously, parameter is difficult to adjust in practical application.
Further, since wheeled mobile robot is typical drive lacking, non-linear object, the rapidity of system response is adjusted
Complicated optimization is needed to calculate, it is difficult to be suitable for real system application.
In conclusion how to design controller in the Trajectory Tracking Control of wheeled mobile robot, guarantee that system is steady
While determining, controller parameter is enabled to have specific adjusting relationship to tracing control performance, to adapt to application request,
It has important theoretical significance and practical application value.
Summary of the invention
It is a primary object of the present invention to overcome drawbacks described above in the prior art, a kind of wheeled mobile robot is proposed
Approximate Decoupling, quick Trajectory Tracking Control method have simple structure, controlled device Approximate Decoupling, adjustment parameter few, convergence speed
Spend the advantages that fast.
The present invention adopts the following technical scheme:
A kind of Approximate Decoupling of wheeled mobile robot, quick Trajectory Tracking Control method, which is characterized in that including as follows
Step: 1) according to the kinematics model of wheeled mobile robot and desired trajectory model, error model and error differential side are established
Journey；2) new control law is used, the forward speed of wheeled mobile robot and the expression formula of steering angular velocity are provided；3) table is set
It is required up to the parameter of formula with meeting decoupling and quickly adjusting.
The error model in step 1) are as follows:
Wherein [xe ye θe]TFor wheeled mobile robot reality
The deviation of pose and expected pose introduces angle, θ in angular errore/ 2 tangent value, can be by θeThe angle of ∈ (- π, π)
It is converted into the convenient value range of (- ∞, ∞), (xr, yr) it is desired locations coordinate of the mass center under world coordinate system, θrFor wheel
The expectation angle of the X-direction of the direction of motion and world coordinate system of formula mobile robot, (x, y) are mass centers under world coordinate system
Actual position coordinate, θ be wheeled mobile robot the direction of motion and world coordinate system X-direction practical angle, θe∈
(- π, π).
The error differential equation in step 1) are as follows:
WhereinV, w is the actual linear velocity of wheeled mobile robot, actual angular speed, v respectivelyr、wrRespectively
It is the expectation linear velocity of wheeled mobile robot, it is expected angular speed,It is x respectivelye、ye, Θ derivative.
V is directed in the step 2)r≠ 0 design the new control law be
In control law vcUnder, haveState x at this timeeIt is unrelated with other states, and work as k1When > 0, xeFast convergence
To 0；And other states y designed in closed-loop systemeWithIt can be directed to v respectively by Lyapunov stability theoryr
> 0 and vr< 0 proves closed-loop system stability, k1、k2、k3It is greater than zero constant.
In the step 2), for vrThe control law of=0 design are as follows:
In control law wcUnder, haveState θ at this timeeIt is unrelated with other states, and work as k3When > 0, θeFast convergence
To 0；And other states x designed in closed-loop systemeAnd ye, closed-loop system can be proved by Lyapunov stability theory
Stability.
In the step 3), the parameter includes parameter k1, k2, k3:
To vr≠ 0 the case where, if parameter meets 0 < < k1, k2, k3, k3< k1, k2|vr| < < k1k3WithThen meet Approximate Decoupling, quick tracking condition；
To vr=0 the case where, if parameter meets 0 < < k1, 0 < < k3,WithThen
Meet Approximate Decoupling, quick tracking condition.
By the above-mentioned description of this invention it is found that compared with prior art, the invention has the following beneficial effects:
A kind of Approximate Decoupling of wheeled mobile robot of the invention, quick Trajectory Tracking Control method, so that wheeled shifting
The pose parameter of mobile robot Approximate Decoupling and can realize quick track following.The contrail tracker proposed has control
The advantages that structure processed is simple, controlled device Approximate Decoupling, adjustment parameter are few, fast convergence rate.
Detailed description of the invention
Fig. 1 is the closed-loop feedback control system schematic diagram of wheeled mobile robot trace tracing control；
Fig. 2 is wheeled mobile robot model schematic；
Fig. 3 is wheeled mobile robot error model schematic diagram；
Fig. 4 is the reference locus curve graph and actual path curve graph of uniform circular motion；
Fig. 5 is the expectation linear velocity curve graph and actual linear velocity curve graph of uniform circular motion；
Fig. 6 is the expectation angular speed curve graph and actual angular speed curve graph of uniform circular motion；
Fig. 7 is the position and attitude error curve graph of the track following of uniform circular motion；
Fig. 8 is the reference locus curve graph and actual path curve graph of speed change cosinusoidal motion；
Fig. 9 is the expectation linear velocity curve graph and actual linear velocity curve graph of speed change cosinusoidal motion；
Figure 10 is the expectation angular speed curve graph and actual angular speed curve graph of speed change cosinusoidal motion；
Figure 11 is the position and attitude error curve graph of the track following of speed change cosinusoidal motion.
Figure 12 is the reference locus curve graph and actual path curve graph for rotating in place movement；
Figure 13 is the expectation linear velocity curve graph and actual linear velocity curve graph for rotating in place movement；
Figure 14 is the expectation angular speed curve graph and actual angular speed curve graph for rotating in place movement；
Figure 15 is the position and attitude error curve graph for rotating in place the track following of movement.
Specific embodiment
Below by way of specific embodiment, the invention will be further described.
A kind of Approximate Decoupling of wheeled mobile robot, quick Trajectory Tracking Control method, so that wheeled mobile robot
Pose parameter Approximate Decoupling and can realize quick track following.The contrail tracker proposed has control structure letter
The advantages that list, controlled device Approximate Decoupling, adjustment parameter are few, fast convergence rate.
Fig. 1 is wheeled mobile robot trace tracing control structural schematic diagram.Inner ring uses conventional speeds controller, realizes
The control of mobile robot space rate and steering angular velocity.Outer ring uses contrail tracker proposed by the present invention, by the phase
The trajectory coordinates x of prestiger、yr、θrIt is compared with feedback x, y, θ, forms deviation xe、ye、θeUsing track proposed by the present invention with
Track control method obtains the space rate v of robotcWith steering angular velocity wc。
Fig. 2 is the model schematic of wheeled mobile robot, and wherein XOY is world coordinate system, and P is wheeled mobile robot
Mass center and geometric center, d be the distance between mass center and geometric center of wheeled mobile robot, l is that two driving wheels are several
The spacing at what center.X, y, θ are used to indicate that 3 space orientation freedom degrees of wheeled mobile robot attained pose, and θ is wheeled
The practical angle of the X-direction of the direction of motion and world coordinate system of mobile robot.vL、vRIt is the left and right wheel of wheeled mobile robot
Actual linear velocity, v, w are the actual linear velocity of wheeled mobile robot, actual angular speed respectively.
Fig. 3 is the error model schematic diagram of wheeled mobile robot, and wherein XOY is world coordinate system, xr、yr、θrBe for
Indicate that wheeled mobile robot refers to 3 space orientation freedom degrees of pose, θrBe wheeled mobile robot the direction of motion with
The reference angle of the X-direction of world coordinate system.vr、wrIt is the reference linear velocity and reference angular velocities of wheeled mobile robot, [xe
ye θe]TIt is the position and attitude error of wheeled mobile robot.
Step 1) establishes the kinematics model of wheeled mobile robot:
Wherein v, w are the actual linear velocity of wheeled mobile robot, actual angular speed respectively, and (x, y) is mass center in the world
Actual position coordinate under coordinate system, θ are the practical folder of the direction of motion of wheeled mobile robot and the X-direction of world coordinate system
Angle, [x y θ]TIt is attained pose of the wheeled mobile robot under global coordinate system.The constraint condition of wheeled mobile robot
It isWherein d is the distance between mass center and geometric center of wheeled mobile robot,It is the derivative of x, y, θ.
The present invention considers that the mass center of robot and geometric center are overlapped, so d=0, i.e. constraint condition areThe constraint condition guarantees that the instantaneous velocity on wheeled mobile robot two-wheeled axis is 0.It can be seen that
Wheeled mobile robot is a kind of typical underactuated control system, i.e. input quantity number is less than output quantity number.Secondly, control
Between amount and controlled volume, there are coupled relations, it is difficult to accurate control.
The present invention proposes a kind of new design method, in design control v and w, introduces new parameter, realizes to [x y
θ]TApproximate Decoupling adjust.Establish desired trajectory model:
Wherein (xr, yr) it is desired locations coordinate of the mass center under world coordinate system, θrFor the movement of wheeled mobile robot
The expectation angle of the X-direction of direction and world coordinate system, [xr yr θr]TIt is the expectation posture of wheeled mobile robot, vr、wrPoint
It is not expectation linear velocity, the expectation angular speed of wheeled mobile robot.
According to kinematics model and desired trajectory model foundation error model:
Wherein [xe ye θe]TFor the deviation of wheeled mobile robot attained pose and expected pose.In angular error,
Introduce angle, θe/ 2 tangent value, can be by θeThe angle of ∈ (- π, π) is converted into the convenient value range of (- ∞, ∞).
Establish the differential equation of error model
WhereinIt is x respectivelye、ye, Θ derivative.
For the differential equation, linear representation is constructed:
Wherein λijFor unknown quantity to be designed.Since above formula is nonlinear equation, λijAleatory variable can be taken.
Step 2) is directed to vr≠ 0 and vr=0 liang of class situation, provides design of control law and stability analysis:
(1) consider vr≠ 0 the case where.
Consider first differential equation, has
Take λ12=λ13=0, λ11=-k1, k at this time1> 0.
Therefore, haveHave at this time
V=yew+vrcosθe+k1xe。
Have on frequency domain,Wherein s is frequency domain complex variable, xeIt (0) is error initial value.Therefore, join
Number meets k1> 0, then error xeAutomatically 0 is converged to, and convergence rate is by k1It determines, k1Bigger, convergence rate is faster.
Consider third equation, has
Take λ31=0, λ33=-k3, k at this time3> 0.Have on frequency domainWherein s is frequency
Domain complex variable,θeIt (0) is error initial value.Therefore, parameter meets k3> 0, then error Θ restrains automatically
It arrivesAnd convergence rate is by k3It determines, k3Bigger, convergence rate is faster.λ32It is undetermined.At this point, having
Consider second equation,
W is brought into obtain
Desirable λ21=-wr-k3sinθe,So that equation is set up.At this point, working as k1> 0k3
> 0, there is xe→ 0,To second equation, approximate analysis can be carried out, is had
In order to make yeConvergence, it is desirable thatDesirable λ32=-k2sign[vr], k2> 0 is then
Based on above-mentioned analysis, from the point of view of rapidity angle, vr, there is following characteristic in ≠ 0 the case where:
1) if there is k3< k1,Then xeCan fast convergence, and to yeWithConvergence it is several
It does not have an impact.At this point, Parameter Conditions can approximation write as: k3< k1And k2|vr| < < k1k3。
2) it when assuming that angular error is in OK range, takesThenIt is capable of fast trackingAt this point, Parameter Conditions are write as:
3)k2If value is sufficiently large, and allows k1And k3Meet above-mentioned condition, then has yeRapidly converge to 0.
Features described above shows under certain condition, set controller parameter k1、k2And k3With to wheel type mobile machine
The separately adjustable characteristic of device people's pose, that is, realize decoupling control.
Finally, vrThe control law of system when ≠ 0 are as follows:
Wherein k1、k2、k3It is greater than zero constant.
In conjunction with the Trajectory Tracking Control rule that above-mentioned analysis obtains, v is providedrThe closed-loop system stability of ≠ 0 situation proves.
In view of having at control law vState x at this timeeIt is unrelated with other states, and work as k1When > 0, xeFastly
Speed converges to 0.Therefore, xeIt will not influence the stability of closed-loop system.Only consider state yeWith?.
Further analyze ye,Influence to stability of control system.It enables
Choose energy equation: V=xTPx, wherein enablingFor symmetric positive definite matrix.To V derivation, obtain:
It is calculated to simplify, takes P12=0.The differential equation of error model and control law are substituted into above formula, obtained:
Since there are sign [vr], to vr> 0 and vrThe case where 0 <, discusses.
1) work as vrWhen > 0,
Wherein,At this point, considering θe∈ (- π, π), in the model
Any value is enclosed, P can be found11> 0, P22> 0 meets
2) work as vrWhen < 0,
Wherein,At this point, considering θe∈ (- π, π), in the model
Any value is enclosed, P can be found11> 0, P22> 0 meets
Therefore, vrWhen ≠ 0, the stability of closed-loop system is proven.
(2) consider vr=0 the case where.
Consider first differential equation, has
Take λ13=0, λ11=-k1, λ12=k2sign[wr] 0 < k at this time1, k2。
Therefore, haveHave at this time
V=yew+vrcosθe+k1xe-k2sign[wr]ye。
Then xeIt converges toConvergence rate is by k1It determines, k1Bigger, convergence rate is faster.
Consider third equation, has
Take λ31=0, λ32=0, λ33=-k3, k at this time3> 0.Haveθ at this timeeAutomatically 0 is converged to, and restrains speed
Degree is by k3It determines, k3Bigger, convergence rate is faster.At this point, having
W=wr+k3sinθe。
Consider second equation,
W is brought into obtainDesirable λ21=-wr-
k3sinθe, λ22=0,So that equation is set up.At this point, working as k1> 0, k3> 0, has θe→ 0, to second equation, approximate analysis can be carried out, is had
Based on above-mentioned analysis, from the point of view of rapidity angle, vr, there is following characteristic in=0 the case where:
1) if there is 0 < < k1, then xeCan fast convergence extremelyAt this point, Parameter Conditions are write as: 0 < <
k1。
2) if there is 0 < < k3, thenOr θeCan fast convergence to 0.At this point, Parameter Conditions are write as: 0 < < k3。
3)AndThen there is ye0 can be converged to.Therefore, Parameter Conditions are write as:
Features described above shows under certain condition, set controller parameter k1、k2And k3With to wheel type mobile machine
The separately adjustable characteristic of device people's pose, that is, realize decoupling control.
Finally, the control law of system are as follows:
Wherein k1、k2、k3It is greater than zero constant.
Then, to vrThe stability that=0 the case where provides closed-loop system proves.
In view of having at control law wState θ at this timeeIt is unrelated with other states, and work as k3
When > 0, θeRapidly converge to 0.Therefore, θeIt will not influence the stability of closed-loop system.Only consider state xeAnd ye?.
Further analyze xe, yeInfluence to stability of control system.Enable x=[xe ye]T
Choose energy equation: V=xTPx, wherein enablingFor symmetric positive definite matrix.To V derivation, obtain:
In above formula, is calculated to simplify, take P12=0.The differential equation of error model and control law are substituted into above formula,
:
Since there are sign [wr], to wr> 0 and wrThe case where 0 <, discusses.
1) work as wrWhen > 0,
Wherein,At this point, P can be found11> 0, P22> 0 meets (P11k2-P22wr)=0.
2) work as wrWhen < 0,
Wherein,At this point, P can be found11> 0, P22> 0 meets (- P11k2-P22wr)=0.
As it can be seen that can find energy equation V meets Lyapunov theorem of stability.Therefore, vrWhen=0, closed-loop system
Stability be proven.
Step 3), according to vr≠ 0 and vr=0, select parameter k1, k2And k3, meet Approximate Decoupling, quick tracking condition,
In to vr, there is 0 < < k in ≠ 0 the case where1, k2, k3, k3< k1, k2|vr| < < k1k3Withvr=0 the case where, there is 0
< < k1, 0 < < k3,
In an embodiment of the present invention, pursuit path uses the following two kinds model:
(1) uniform circular motion, wherein vr=0.2m/s, wr=0.2rad/s.Assuming that the velocity original value of system is v0=
0.1m/s, w0=0.1rad/s is set as with reference to the initial value of poseThe initial value of attained pose is set as
The parameter chosen at this time is k1=20, k2=35, k3=12.Fig. 4 be uniform circular motion reference locus curve graph and practical rail
Trace curve figure, Fig. 5 are the expectation linear velocity curve graph and actual linear velocity curve graph of uniform circular motion, and Fig. 6 is steady circular
The expectation angular speed curve graph and actual angular speed curve graph of movement, Fig. 7 are that the pose of the track following of uniform circular motion misses
Dygoram.
(2) speed change cosinusoidal motion, whereinAssuming that the speed of system
Degree initial value is v0=0.1m/s, w0=0.1rad/s is set as with reference to the initial value of poseThe initial value of attained pose
It is set asThe parameter chosen at this time is k1=20, k2=35, k3=12.Fig. 8 is the reference locus of speed change cosinusoidal motion
Curve graph and actual path curve graph, Fig. 9 are the expectation linear velocity curve graph and actual linear velocity curve graph of speed change cosinusoidal motion,
Figure 10 is the expectation angular speed curve graph and actual angular speed curve graph of speed change cosinusoidal motion, and Figure 11 is the rail of speed change cosinusoidal motion
The position and attitude error curve graph of mark tracking.
(3) movement is rotated in place, wherein vr=0, wr=sin t rad/s.Assuming that the velocity original value of system is v0=
0.1m/s, w0=0.1rad/s is set as with reference to the initial value of poseThe initial value of attained pose is set as
The parameter chosen at this time is k1=20, k2=35, k3=12.Figure 12 is the reference locus curve graph and reality for rotating in place movement
Geometric locus figure, Figure 13 are the expectation linear velocity curve graph and actual linear velocity curve graph for rotating in place movement, and Figure 14 is original place
The expectation angular speed curve graph and actual angular speed curve graph of rotary motion, Figure 15 is the position for rotating in place the track following of movement
Appearance error curve diagram.
The above is only a specific embodiment of the present invention, but the design concept of the present invention is not limited to this, all to utilize this
Design makes a non-material change to the present invention, and should all belong to behavior that violates the scope of protection of the present invention.
Claims (6)
1. a kind of Approximate Decoupling of wheeled mobile robot, quick Trajectory Tracking Control method, which is characterized in that including walking as follows
It is rapid: 1) according to the kinematics model of wheeled mobile robot and desired trajectory model, to establish error model and the error differential equation；
2) new control law is used, the forward speed of wheeled mobile robot and the expression formula of steering angular velocity are provided；3) setting expression
The parameter of formula is required with meeting decoupling and quickly adjusting.
2. a kind of Approximate Decoupling of wheeled mobile robot as described in claim 1, quick Trajectory Tracking Control method, special
Sign is, the error model in step 1) are as follows:
Wherein [xe ye θe]TAngle is introduced in angular error for the deviation of wheeled mobile robot attained pose and expected pose
Spend θe/ 2 tangent value, can be by θeThe angle of ∈ (- π, π) is converted into the convenient value range of (- ∞, ∞), (xr,yr) it is matter
Desired locations coordinate of the heart under world coordinate system, θrFor the direction of motion of wheeled mobile robot and the side X of world coordinate system
To expectation angle, (x, y) is actual position coordinate of the mass center under world coordinate system, and θ is the movement of wheeled mobile robot
The practical angle of the X-direction of direction and world coordinate system, θe∈(-π,π)。
3. a kind of Approximate Decoupling of wheeled mobile robot as claimed in claim 2, quick Trajectory Tracking Control method, special
Sign is, the error differential equation in step 1) are as follows:
WhereinV, w is the actual linear velocity of wheeled mobile robot, actual angular speed, v respectivelyr、wrIt is wheel respectively
The expectation linear velocity of formula mobile robot, expectation angular speed,It is x respectivelye、ye, Θ derivative.
4. a kind of Approximate Decoupling of wheeled mobile robot as claimed in claim 3, quick Trajectory Tracking Control method, special
Sign is, v is directed in the step 2)r≠ 0 design the new control law be
At control law vc, haveState x at this timeeIt is unrelated with other states, and work as k1When > 0, xeIt rapidly converges to
0；And other states y designed in closed-loop systemeWithIt can be directed to v respectively by Lyapunov stability theoryr
> 0 and vr< 0 proves closed-loop system stability, k1、k2、k3It is greater than zero constant.
5. a kind of Approximate Decoupling of wheeled mobile robot as claimed in claim 3, quick Trajectory Tracking Control method, special
Sign is, in the step 2), for vrThe control law of=0 design are as follows:
At control law wc, haveState θ at this timeeIt is unrelated with other states, and work as k3When > 0, θeIt rapidly converges to
0；And other states x designed in closed-loop systemeAnd ye, can prove that closed-loop system is steady by Lyapunov stability theory
It is qualitative.
6. a kind of Approximate Decoupling of wheeled mobile robot as claimed in claim 3, quick Trajectory Tracking Control method, special
Sign is, in the step 3), the parameter includes parameter k1, k2, k3:
To vr≠ 0 the case where, if parameter meets 0 < < k1,k2,k3, k3< k1, k2|vr| < < k1k3And 2k2|vr| < < k3 2
Then meet Approximate Decoupling, quick tracking condition；
To vr=0 the case where, if parameter meets 0 < < k1, 0 < < k3,WithThen meet
Approximate Decoupling, quick tracking condition.
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| CN111830994A (en) * | 2020-08-05 | 2020-10-27 | 华侨大学 | Wheel type mobile robot speed control method and system |
| CN111830994B (en) * | 2020-08-05 | 2022-04-01 | 华侨大学 | Wheel type mobile robot speed control method and system |
| CN112068426A (en) * | 2020-08-26 | 2020-12-11 | 广州大学 | Wheeled robot finite time control method, system, device and medium |
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