CN108107890B - A kind of safe trajectory planing method of Nonlinear Uncertain Systems - Google Patents
A kind of safe trajectory planing method of Nonlinear Uncertain Systems Download PDFInfo
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- G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
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- G05D1/0214—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory in accordance with safety or protection criteria, e.g. avoiding hazardous areas
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- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
- G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
- G05D1/02—Control of position or course in two dimensions
- G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
- G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
- G05D1/0221—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory involving a learning process
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Abstract
The invention discloses a kind of safe trajectory planing methods of Nonlinear Uncertain Systems, the present invention is a kind of new nonlinear kinetics paths planning method, consider measurement and systematic error, it proposes and local feedback control invariant set size is corrected using the method for huge Baudrillard gold collection difference again, and carry out the algorithm of path planning online.Specific steps include: Step 1: open loop track library calculates；Step 2: calculating local invariant collection；Step 3: calculating error propagation；Step 4: path planning.From simulation result it can be seen that the validity of method proposed by the invention and algorithm, the present invention can form the secure path of Global robust, to meet the requirement of Nonlinear Uncertain Systems safe trajectory planning.
Description
Technical field
The present invention relates to nonlinear system path planning and dynamics Controlling field, in particular to a kind of nonlinear uncertain
The safe trajectory planing method of system.
Background technique
If robot is cannot carry out path planning in the case where oneself fully known state, using algorithm for estimating come pre-
Possible path is surveyed to find safety and the lesser path of error, such as BRM (belief roadmap) algorithm, RRBT
(Rapidly-exploring Random Belief Trees) algorithm.Compared to traditional routing algorithm, consider it is uncertain because
Element provides a good guarantee for safe path, avoids because probabilistic have what influence physical planning path was run
Safety.But dynamic system does not ensure that the operation of safe trajectory, such as executive capability are saturated sometimes, system deviation etc.,
Therefore it has been born and has met the path planning algorithms of Dynamic Constraints such as.And for nonlinear system, meet linear control method
Dynamic Prediction can be only present in the neighborhood of very little, Russ professor Tedrake in the laboratory EECS of MIT proposes to use
LQR-trees method is converted the local linear feedback control overall situation for nonlinear Control problem and is connected using random search tree algorithm
The form connect forms the suboptimum track for meeting the requirement of dynamics feedback control, this is a kind of nonlinear control algorithm.Domain of attraction
The feasibility problems of local linearization are solved perfectly with the introducing of invariant set, Existence of Global Stable is completed in hybrid system switching.It is non-thread
Property system attractive domain and invariant set calculating it is very time-consuming, therefore A.Majumda introduces path library and carries out off-line calculation, and adopts
Combined sequence method case in carries out merging the connection for completing invariant set online.
Summary of the invention
In order to overcome disadvantage mentioned above, the present invention provides a kind of new nonlinear kinetics paths planning method, this method
Consider feedback path planning in existing measuring uncertainty, and in the case that error there are feedback control still can be stable arrive
Up to dbjective state.
The present invention is to be achieved through the following technical solutions:
A kind of safe trajectory planing method of Nonlinear Uncertain Systems, comprising the following steps:
Step 1: open loop track library calculates
Path point is generated using PRM method at random, designs attachable kinetic locus for adjacent path point line, it is raw
It is nonlinear programming problem at local path library, and carries out that open loop track and opened loop control, open loop track and open loop is calculated
Control constitutes track library；
Step 2: calculating local invariant collection
Off-line calculation is carried out using the algorithm of nonlinear system domain of attraction and invariant set, is obtained for the peace for determining reachable set
The range of full stability-of-path；
Step 3: calculating error propagation
First by Nonlinear Systems ' Discrete, partial feedback coefficient K is obtained by LQR algorithmkControl rate is obtained, is missed
Poor elliptic equation；
Step 4: path planning
Pipeline is in tkValue E (the t at momentk) and error ellipse ε (0, Pk) collection difference operation is carried out, it obtains credible under error condition
Pipeline, and the logical estimated value provided detects whether the inlet that true value section can be allowed be in next pipeline composition sequence
Connection, and complete path planning splicing.
In step 2, off-line calculation is carried out using SOS algorithm, for all tracks in step 1 in the library of track, is solved
Time-varying Riccati equation obtains the lyapunov function in time interval, the initial value as SOS algorithm；And then it obtains for true
Determine the stable range of the safe trajectory of reachable set.
In step 2, time-varying Riccati equation are as follows:
Wherein S (t) is matrix to be solved, and A (t) is the later coefficient matrix of system linearization, and Q, R are design parameter square
Battle array.
Error ellipse equation specific steps are obtained in step 3 are as follows:
Definition: oval ε (q, Q) indicates that q is oval center of circle phasor coordinate, and Q is elliptic parameter, is poised for battle matrix for positive definite.
And(<>indicates inner product)；
First by Nonlinear Systems ' Discrete:
xk+1=g (xk,uk,wk)
yk+1=h (xk,vk)
Wherein, wkAnd vk+1Respectively system noise and observation noise.X and y is respectively state variable and observed quantity variable, u
For system input.Partial feedback coefficient K is obtained by LQR liner quadratic regulator algorithmkObtain control rate:
Assuming thatFor the error of state actual value and estimated value, error ellipse:
ee∈ε(0,Pk)
PkFor posteriori error estimate, it is assumed thatFor the error of state estimation and nominal state,
Wherein AkFor system linearization coefficient, HkFor the Jacobian matrix of observational equation.Lk+1For filtering gain.It enablesVirtual condition indicates are as follows:
So that
Wherein: AG=Ak+BkKk, Therefore the predicted value nominally estimated:Wherein elliptic parameter are as follows:
In step 4, specific algorithm is as follows:
V, n indicate that random walk node, Q indicate that data structure storehouse, v.path indicate to reach the path of present node, road
Diameter by dynamics estimation be attached, be stored in v [MT], MT be a tuple { τ, F, c, u } respectively indicate track,
Funnel, consuming value, control force }, FORECAST (n, v) is the calculating of two node error propagations, obtains new contain not really
Qualitative funnel F, updates the value in v [MT], SEQCOMPOSITION (F1,F2) indicate two funnel sequence connection meters
It calculates, can connect return 1 otherwise is 0.
Compared with prior art, the invention has the following beneficial technical effects:
The present invention considers influence of the measuring uncertainty for feedback control stability region, is contracted using huge Baudrillard gold collection difference
The stabilization of system, reappraises path using estimation of deviation in the case of the size in small practical stability region exists with error in judgement
Safe avoidance problem in planning process, and optimal trajectory is searched for using path search algorithm.Specifically, using off-line calculation
Mode of the motor pool in splicing and detection, road of the design nonlinear motion robot under constraint and measurement uncertain condition
Diameter forms a kind of safe trajectory planing method of Nonlinear Uncertain Systems.This method is existing in considering feedback path planning
Measuring uncertainty, and in the case that error there are feedback control still can be stable arrival dbjective state.It can from simulation result
To find out that the validity of method proposed by the invention and algorithm, the present invention can form the secure path of Global robust, thus full
The requirement of sufficient Nonlinear Uncertain Systems safe trajectory planning.
Detailed description of the invention
Fig. 1 is path planning graph and kinetic locus；
Fig. 2 is PRM node and most short kinetic pathways；
Fig. 3 is most short safe trajectory；
Fig. 4 is safe trajectory when error increases；
Fig. 5 is the safe trajectory when local measurement missing.
Specific embodiment
With reference to the accompanying drawing and specific embodiment the present invention will be further described.
The present invention by path planning measuring uncertainty and newest dynamics program results combine, propose one kind
Feedback path based on probabilistic roadmap method plans (FBRM, Feedback Belief Roadmap), using off-line calculation motor pool
In the mode of splicing and detection, designs nonlinear motion robot and constraining and measuring the path under uncertain condition, shape
At a kind of safe trajectory planing method of Nonlinear Uncertain Systems.Consider measurement and systematic error, proposes using Pang Deli
The method of sub- gold collection difference corrects local feedback control invariant set size again, and carries out the algorithm of path planning online.This method
Including following four step:
Step 1: open loop track library calculates
Path point is generated using PRM method at random, designs attachable dynamics for adjacent path point line (edge)
Track.Generation local path library is nonlinear programming problem (NLP, nonlinear programming), soft using GPOPS-II
Part carries out that open loop track and opened loop control is calculated, these tracks and control constitute track library.As shown in Figure 1.
Step 2: calculating local invariant collection
There are many algorithm of nonlinear system domain of attraction and invariant set, and the present invention uses SOS (sums of squares) algorithm
Carry out off-line calculation.Firstly, solving time-varying Riccati equation for all tracks in track library in step 1:Obtain the lyapunov function S in time interval t ∈ [0 T]
(t), the initial value as SOS algorithm.SOS algorithm is available for the stable range of the safe trajectory of reachable set is determined, this can
Be expressed as a pipeline (funnel) with image, into funnel stateful be proved to that specific objective collection can be reached
State.The corresponding corresponding funnel in track in track library each in this way.
Step 3: calculating error propagation
Definition: oval ε (q, Q) indicates that q is the oval center of circle, and(<>table
Show inner product).
First by Nonlinear Systems ' Discrete:
Wherein wkAnd vk+1Respectively system noise and observation noise.Partial feedback coefficient K is obtained by LQR algorithmkIt obtains
Control rate:
Assuming thatFor the error of actual value and estimated value, error ellipse:
ee∈ε(0,Pk) (3)
PkFor posteriori error estimate, ε () indicates oval.Posterior estimator error can be using ESMF (extension set-membership filtering)
Or EKF (Extended Kalman filter) and other filtering algorithms obtain.Assuming thatFor optimal estimation and nominal state
Error.
It enablesSo virtual condition can indicate are as follows:
So that
Wherein: AG=Ak+BkKk, Therefore the predicted value nominally estimated:Wherein elliptic parameter:
Step 4: path planning
Funnel is in tkValue E (the t at momentk) and error ellipse ε (0, Pk) carry out the poor (Pontryagin of collection
Difference) operation obtains believable funnel under error condition, and being detected whether by the estimated value of (7) offer can be with
The inlet for allowing true value section to be in next funnel constitutes sequence connection (Sequential composition), and complete
Splice at path planning.Specific algorithm is as follows:
V, n indicate that random walk node, Q indicate that data structure storehouse, v.path indicate to reach the path of present node, this
A little paths are attached by dynamics estimation, are stored in v [MT], and MT is that a tuple { τ, F, c, u } respectively indicates { rail
Mark, funnel, consuming value, control force }, FORECAST (n, v) is the calculating (step 3) of two node error propagations, is obtained new
Contain probabilistic funnel F, update v [MT] in value.SEQCOMPOSITION(F1,F2) indicate two funnel sequences
Column connection calculates, and can connect return 1 otherwise is 0.
By taking ground mobile robot as an example.Assuming that the forward movement speed of ground robot specifications 10m/s, is turned by adjusting
To allowing robot to carry out safe avoidance, as shown in Figure 2-5:
As shown in Figure 2-5, black region is constraint, and gray area funnel, dotted line is the edge of connecting node,
Pore is node, and black fine line is kinetic locus, and yellow line indicates previous funnel exit region, under green ellipse representation
A funnel entrance.Fig. 2 is shown in the map there are barrier in the case where not considering error along kinetic locus
Shortest path, and motion profile will appear deviation in the case where there are noise, therefore in the case where no preferably measurement
This track may stabilization may also rapid divergence to it cannot be guaranteed that safety.It is available very small in error by this method
In the case where run along most short safe trajectory, and be constantly in funnel because of the presence of feedback control, such as Fig. 3 institute
Show.
In the biggish situation of initial error, for shortest path because not measuring, error ellipse has exceeded funnel
Constraint, obtain uncertain state by collecting difference and cannot keep in funnel.Therefore path planning side proposed by the present invention
Method gives the shortest path of new guarantee safety, as shown in Figure 4.Because this paths is there are measuring node dark color dot, from
And guarantees that error is limited in funnel and ensure that safety.But if remove one of measurement point as shown in figure 5, because surveying
Amount can not reduce uncertainty in original route, therefore intermediate path can not ensure that control is stablized, therefore also not can guarantee rail
Mark safety, as shown in figure 5, the track that method of the invention will select another measurement more accurate but distant.From contracting
It can be seen that the deviation ellipse of estimation has been over the inlet radius containing probabilistic new funnel, therefore nothing in sketch map
Method reaches target trajectory.
From simulation result it can be seen that the validity of method proposed by the invention and algorithm.The present invention can form global Shandong
The secure path of stick, to meet the requirement of Nonlinear Uncertain Systems safe trajectory planning.
More than, only presently preferred embodiments of the present invention is not limited only to practical range of the invention, all according to the invention patent
The equivalence changes and modification that the content of range is done all should be technology scope of the invention.
Claims (2)
1. a kind of safe trajectory planing method of Nonlinear Uncertain Systems, which comprises the following steps:
Step 1: open loop track library calculates
Path point is generated using PRM method at random, designs attachable kinetic locus, generation office for adjacent path point line
Portion, library, track is nonlinear programming problem, and carries out that open loop track and opened loop control, open loop track and opened loop control is calculated
Constitute track library；
Step 2: calculating local invariant collection
Off-line calculation is carried out using the algorithm of nonlinear system domain of attraction and invariant set, is obtained for the safe rail for determining reachable set
The stable range of mark；
Step 3: calculating error propagation
First by Nonlinear Systems ' Discrete, partial feedback coefficient K is obtained by LQR algorithmkControl rate is obtained, it is ellipse to obtain error
Equation of a circle；
Step 4: path planning
Pipeline is in tkValue E (the t at momentk) and error ellipse ε (0, Pk) collection difference operation is carried out, obtain believable pipe under error condition
Road, and the logical estimated value provided detects whether that the inlet that true value section can be allowed to be in next pipeline constitutes sequence and connects
It connects, and completes path planning splicing；
In step 2, off-line calculation is carried out using SOS algorithm, for all tracks in step 1 in the library of track, solves time-varying
Riccati equation obtains the lyapunov function in time interval, the initial value as SOS algorithm；Obtaining in turn can for determination
The range stable up to the safe trajectory of collection；
In step 2, time-varying Riccati equation are as follows:
Wherein S (t) is matrix to be solved, and A (t) is the later coefficient matrix of system linearization, and Q, R are design parameter matrix；
Error ellipse equation specific steps are obtained in step 3 are as follows:
Definition: oval ε (q, Q) indicates that q is oval center of circle phasor coordinate, and Q is elliptic parameter, is poised for battle matrix for positive definite, and<>indicates inner product；
First by Nonlinear Systems ' Discrete:
xk+1=g (xk,uk,wk)
yk+1=h (xk,vk)
Wherein, wkAnd vk+1Respectively system noise and observation noise, x and y are respectively state variable and observed quantity variable, and u is to be
System input, obtains partial feedback coefficient K by LQR liner quadratic regulator algorithmkObtain control rate:
Assuming thatFor the error of state actual value and estimated value, error ellipse:
ee∈ε(0,Pk)
PkFor posteriori error estimate, it is assumed thatFor the error of state estimation and nominal state,
Wherein AkFor system linearization coefficient, HkFor the Jacobian matrix of observational equation, Lk+1For filtering gain；It enablesVirtual condition indicates are as follows:
So that
Wherein: AG=Ak+BkKk,
Therefore the predicted value nominally estimated:Wherein elliptic parameter are as follows:
2. a kind of safe trajectory planing method of Nonlinear Uncertain Systems according to claim 1, which is characterized in that step
In rapid four, specific algorithm is as follows:
V, n indicate that random walk node, Q indicate that data structure storehouse, v.path indicate to reach the path of present node, and path is logical
Cross dynamics estimation be attached, be stored in v [MT], MT be a tuple { τ, F, c, u } respectively indicate track, funnel,
Consuming value, control force }, FORECAST (n, v) is the calculating of two node error propagations, is obtained new containing probabilistic
Funnel F updates the value in v [MT], SEQCOMPOSITION (F1,F2) indicate that two funnel sequence connections calculate, it can connect
Connecing return 1 otherwise is 0.
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