From f3a985940eddfe1499831ca2095053b32713d61f Mon Sep 17 00:00:00 2001 From: mcarfagno Date: Mon, 6 Jul 2020 16:54:22 +0100 Subject: [PATCH] f10 pybullet demo --- README.md | 21 ++-- img/demo.gif | Bin 1686691 -> 3430694 bytes img/demo.gif.old | Bin 0 -> 1686691 bytes img/f10.png | Bin 0 -> 55267 bytes img/quicklatex1.png | Bin 4163 -> 14596 bytes img/quicklatex2.png | Bin 1235 -> 2532 bytes mpc_demo/cvxpy_mpc.py | 20 ++-- mpc_demo/mpc_demo_nosim.py | 10 +- mpc_demo/mpc_demo_pybullet.py | 90 ++++++++++++----- mpc_demo/utils.py | 42 +++++--- notebooks/equations.ipynb | 182 ++++++++++++++++++++++++++++------ 11 files changed, 274 insertions(+), 91 deletions(-) create mode 100644 img/demo.gif.old create mode 100644 img/f10.png diff --git a/README.md b/README.md index 2c5f930..36418b6 100644 --- a/README.md +++ b/README.md @@ -4,30 +4,35 @@ Python implementation of a mpc controller for path tracking using **[CVXPY](http ## About -The MPC is a model predictive path following controller which does follow a predefined reference path Xref and Yref by solving an optimization problem. The resulting optimization problem is shown in the following equation: +The MPC is a model predictive path following controller which does follow a predefined reference by solving an optimization problem. The resulting optimization problem is shown in the following equation: ![](img/quicklatex1.png) -The vehicle dynamics are described by the differential drive model: +The terns of the cost function are the sum of the **cross-track error**, **heading error**, **velocity error** and **actuaction effort**. + +Where R,P,K,Q are the cost matrices used to tune the response. + +The vehicle model is described by the bicycle kinematics model using the state space matrices A and B: ![](img/quicklatex2.png) -The state variables of the model are: +The state variables **(x)** of the model are: * **x** coordinate of the robot * **y** coordinate of the robot +* **v** velocuty of the robot * **theta** heading of the robot -The inputs of the model are: +The inputs **(u)** of the model are: -* **v** linear velocity of the robot -* **w** angular velocity of the robot +* **a** linear acceleration of the robot +* **delta** steering angle of the robot ## Demo -The MPC implementation is tested using **[bullet](https://pybullet.org/wordpress/)** physics simulator. Turtlebot model is from: *https://github.com/erwincoumans/pybullet_robots*. +The MPC implementation is tested using **[bullet](https://pybullet.org/wordpress/)** physics simulator. Racing car model is from: *https://github.com/erwincoumans/pybullet_robots*. -![](img/Turtlebot.png) +![](img/f10.png) Results: diff --git a/img/demo.gif b/img/demo.gif index 85c4aa07d3dd8ffaa78b4d10b367f3dd98a02bd4..0a1ea2ed781b1fa531b6ac318bfa744434332a8c 100644 GIT binary patch literal 3430694 zcmeF1`9Bn1^#5l!7>s>q?Au`MOJR(CD`ZP(>=d%5q>Nc?iBVZ9)!5=KG}a_#j5Sh7 z5-R)FVokEmx6kL__>-x#wIPJ8MHDe=pE0;2!`6zybh)Km-CI zARr(qDXA1+uA-u%exps7_@9A+fnip=+2sN(xe}MvZfR-ha5>K*tIi?or6cja)7=gi z7Z;B^79QoA9@Tc<-rnc(I?m;b`V^}8)cN?PJ@LDoasK>yd|r+J{r-T2gutYPK;o^y z;unDrJA(d^9v@s>8`2UT`j7I#aI(vV3m2oJqN1at<6>hI;^PzlQFZ#t9jTE!z1wEEuke>BynH($*xkIQ;`F*`f^7V*NZ z^rYO}+`PQJf`a_}S3DmyzA3&XUR?a(VaJ=tB`!458CvH=Nl8g*qHTH4qbHS>HHii_ zH8szkJ)_st>3zNR4fPFgJ~uWtHaEAljI2C=@v^nGwXLn~ReO6!M@LszSML?w-qESP z{=WV<{ey+!gM)A1POlAR+6|}a4G#~Gz9{)X7W*(ZHkK$cK2$rt@O|QW#^<_-&*M{{ z|1mlF`RnY|=&PyEV^d3CrAG(%>A#;&Cky*EY2+~&n?c*FD%S2EY1JxEY8m_{aBcr zU-&oqZ|r|qU7Y(jytKG5zxcl_eR;REIJdO=Yx&=n3roujYs>%G*jf3d%HYzyV`fVCT!>NnVnLfot>TE 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+ MAX_SPEED = ref_vel*1.5 + MAX_STEER = np.pi/4 MAX_ACC = 1.0 # compute polynomial coefficients of the track @@ -77,9 +77,9 @@ def optimize(state,u_bar,track,ref_vel=1.): for t in range(P.T): - cost += 30*cp.sum_squares(x[3,t]-np.arctan(df(x_bar[0,t],K))) # psi - cost += 20*cp.sum_squares(f(x_bar[0,t],K)-x[1,t]) # cte - cost += 10*cp.sum_squares(ref_vel-x[2,t]) # desired v + cost += 20*cp.sum_squares(x[3,t]-np.clip(np.arctan(df(x_bar[0,t],K)),-np.pi,np.pi) ) # psi + cost += 40*cp.sum_squares(f(x_bar[0,t],K)-x[1,t]) # cte + cost += 20*cp.sum_squares(ref_vel-x[2,t]) # desired v # Actuation rate of change if t < (P.T - 1): @@ -101,10 +101,14 @@ def optimize(state,u_bar,track,ref_vel=1.): # Solve prob = cp.Problem(cp.Minimize(cost), constr) - solution = prob.solve(solver=cp.OSQP, verbose=False) + prob.solve(solver=cp.OSQP, verbose=False) + + if "optimal" not in prob.status: + print("WARN: No optimal solution") + return u_bar #retrieved optimized U and assign to u_bar to linearize in next step - u_bar=np.vstack((np.array(u.value[0, :]).flatten(), + u_opt=np.vstack((np.array(u.value[0, :]).flatten(), (np.array(u.value[1, :]).flatten()))) - return u_bar + return u_opt diff --git a/mpc_demo/mpc_demo_nosim.py b/mpc_demo/mpc_demo_nosim.py index cf24f90..0b00533 100755 --- a/mpc_demo/mpc_demo_nosim.py +++ b/mpc_demo/mpc_demo_nosim.py @@ -34,8 +34,8 @@ class MPC(): # Interpolated Path to follow given waypoints #self.path = compute_path_from_wp([0,10,12,2,4,14],[0,0,2,10,12,12]) - self.path = compute_path_from_wp([0,3,4,6,10,12,14,6,1,0], - [0,0,2,4,3,3,-2,-6,-2,-2],1) + self.path = compute_path_from_wp([0,3,4,6,10,12,13,13,6,1,0], + [0,0,2,4,3,3,-1,-2,-6,-2,-2],0.5) # Sim help vars self.sim_time=0 @@ -174,7 +174,7 @@ class MPC(): plt.subplot(grid[1, 2]) #plt.title("Angular Velocity {} m/s".format(self.w_history[-1])) plt.plot(np.degrees(self.d_history),c='tab:orange') - plt.ylabel('w(t) [deg]') + plt.ylabel('gamma(t) [deg]') locs, _ = plt.xticks() plt.xticks(locs[1:], locs[1:]*P.dt) plt.xlabel('t [s]') @@ -186,8 +186,8 @@ class MPC(): def plot_car(x, y, yaw): - LENGTH = 0.3 # [m] - WIDTH = 0.1 # [m] + LENGTH = 0.35 # [m] + WIDTH = 0.2 # [m] OFFSET = LENGTH # [m] outline = np.array([[-OFFSET, (LENGTH - OFFSET), (LENGTH - OFFSET), -OFFSET, -OFFSET], diff --git a/mpc_demo/mpc_demo_pybullet.py b/mpc_demo/mpc_demo_pybullet.py index 910bf88..123b895 100644 --- a/mpc_demo/mpc_demo_pybullet.py +++ b/mpc_demo/mpc_demo_pybullet.py @@ -20,21 +20,26 @@ def get_state(robotId): robPos, robOrn = p.getBasePositionAndOrientation(robotId) linVel,angVel = p.getBaseVelocity(robotId) - return[robPos[0], robPos[1], p.getEulerFromQuaternion(robOrn)[2]] + return[robPos[0], robPos[1], linVel[0], p.getEulerFromQuaternion(robOrn)[2]] -def set_ctrl(robotId,v,w): - """ - """ - L= 0.354 - R= 0.076/2 +def set_ctrl(robotId,currVel,acceleration,steeringAngle): - rightWheelVelocity= (2*v+w*L)/(2*R) - leftWheelVelocity = (2*v-w*L)/(2*R) + gearRatio=1./21 + steering = [0,2] + wheels = [8,15] + maxForce = 50 - p.setJointMotorControl2(robotId,0,p.VELOCITY_CONTROL,targetVelocity=leftWheelVelocity,force=1000) - p.setJointMotorControl2(robotId,1,p.VELOCITY_CONTROL,targetVelocity=rightWheelVelocity,force=1000) + targetVelocity = currVel + acceleration*P.dt + #targetVelocity=lastVel + #print(targetVelocity) -def plot(path,x_history,y_history): + for wheel in wheels: + p.setJointMotorControl2(robotId,wheel,p.VELOCITY_CONTROL,targetVelocity=targetVelocity/gearRatio,force=maxForce) + + for steer in steering: + p.setJointMotorControl2(robotId,steer,p.POSITION_CONTROL,targetPosition=steeringAngle) + +def plot_results(path,x_history,y_history): """ """ plt.style.use("ggplot") @@ -52,24 +57,54 @@ def run_sim(): """ p.connect(p.GUI) - start_offset = [0,2,0] - start_orientation = p.getQuaternionFromEuler([0,0,0]) - turtle = p.loadURDF("turtlebot.urdf",start_offset, start_orientation) - plane = p.loadURDF("plane.urdf") + p.resetSimulation() - p.setRealTimeSimulation(1) p.setGravity(0,0,-10) + useRealTimeSim = 1 - # MPC time step - P.dt = 0.25 + p.setTimeStep(1./120.) + p.setRealTimeSimulation(useRealTimeSim) # either this + + plane = p.loadURDF("racecar/plane.urdf") + #track = p.loadSDF("racecar/f10_racecar/meshes/barca_track.sdf", globalScaling=1) + + car = p.loadURDF("racecar/f10_racecar/racecar_differential.urdf", [0,0,.3]) + for wheel in range(p.getNumJoints(car)): + print("joint[",wheel,"]=", p.getJointInfo(car,wheel)) + p.setJointMotorControl2(car,wheel,p.VELOCITY_CONTROL,targetVelocity=0,force=0) + p.getJointInfo(car,wheel) + + c = p.createConstraint(car,9,car,11,jointType=p.JOINT_GEAR,jointAxis =[0,1,0],parentFramePosition=[0,0,0],childFramePosition=[0,0,0]) + p.changeConstraint(c,gearRatio=1, maxForce=10000) + + c = p.createConstraint(car,10,car,13,jointType=p.JOINT_GEAR,jointAxis =[0,1,0],parentFramePosition=[0,0,0],childFramePosition=[0,0,0]) + p.changeConstraint(c,gearRatio=-1, maxForce=10000) + + c = p.createConstraint(car,9,car,13,jointType=p.JOINT_GEAR,jointAxis =[0,1,0],parentFramePosition=[0,0,0],childFramePosition=[0,0,0]) + p.changeConstraint(c,gearRatio=-1, maxForce=10000) + + c = p.createConstraint(car,16,car,18,jointType=p.JOINT_GEAR,jointAxis =[0,1,0],parentFramePosition=[0,0,0],childFramePosition=[0,0,0]) + p.changeConstraint(c,gearRatio=1, maxForce=10000) + + + c = p.createConstraint(car,16,car,19,jointType=p.JOINT_GEAR,jointAxis =[0,1,0],parentFramePosition=[0,0,0],childFramePosition=[0,0,0]) + p.changeConstraint(c,gearRatio=-1, maxForce=10000) + + c = p.createConstraint(car,17,car,19,jointType=p.JOINT_GEAR,jointAxis =[0,1,0],parentFramePosition=[0,0,0],childFramePosition=[0,0,0]) + p.changeConstraint(c,gearRatio=-1, maxForce=10000) + + c = p.createConstraint(car,1,car,18,jointType=p.JOINT_GEAR,jointAxis =[0,1,0],parentFramePosition=[0,0,0],childFramePosition=[0,0,0]) + p.changeConstraint(c,gearRatio=-1, gearAuxLink = 15, maxForce=10000) + c = p.createConstraint(car,3,car,19,jointType=p.JOINT_GEAR,jointAxis =[0,1,0],parentFramePosition=[0,0,0],childFramePosition=[0,0,0]) + p.changeConstraint(c,gearRatio=-1, gearAuxLink = 15,maxForce=10000) opt_u = np.zeros((P.M,P.T)) - opt_u[0,:] = 1 #m/s - opt_u[1,:] = np.radians(0) #rad/s + opt_u[0,:] = 1 #m/ss + opt_u[1,:] = np.radians(0) #rad/ # Interpolated Path to follow given waypoints - path = compute_path_from_wp([0,3,4,6,10,13], - [0,0,2,4,3,3],1) + path = compute_path_from_wp([0,3,4,6,10,11,12,6,1,0], + [0,0,2,4,3,3,-1,-6,-2,-2],0.5) for x_,y_ in zip(path[0,:],path[1,:]): p.addUserDebugLine([x_,y_,0],[x_,y_,0.33],[0,0,1]) @@ -77,9 +112,10 @@ def run_sim(): x_history=[] y_history=[] + time.sleep(0.5) while (1): - state = get_state(turtle) + state = get_state(car) x_history.append(state[0]) y_history.append(state[1]) @@ -88,18 +124,18 @@ def run_sim(): if np.sqrt((state[0]-path[0,-1])**2+(state[1]-path[1,-1])**2)<1: print("Success! Goal Reached") - set_ctrl(turtle,0,0) - plot(path,x_history,y_history) + set_ctrl(car,0,0,0) + plot_results(path,x_history,y_history) p.disconnect() return #optimization loop start=time.time() - opt_u = optimize(state,opt_u,path) + opt_u = optimize(state,opt_u,path,ref_vel=1.0) elapsed=time.time()-start print("CVXPY Optimization Time: {:.4f}s".format(elapsed)) - set_ctrl(turtle,opt_u[0,1],opt_u[1,1]) + set_ctrl(car,state[2],opt_u[0,1],opt_u[1,1]) if P.dt-elapsed>0: time.sleep(P.dt-elapsed) diff --git a/mpc_demo/utils.py b/mpc_demo/utils.py index 03b6a9f..330b822 100755 --- a/mpc_demo/utils.py +++ b/mpc_demo/utils.py @@ -9,9 +9,8 @@ def compute_path_from_wp(start_xp, start_yp, step = 0.1): delta = step #[m] for idx in range(len(start_xp)-1): - section_len = np.sum(np.sqrt(np.power(np.diff(start_xp[idx:idx+2]),2)+np.power(np.diff(start_yp[idx:idx+2]),2))) - - interp_range = np.linspace(0,1,np.floor(section_len/delta).astype(int)) + section_len = np.sqrt(np.sum(np.power(np.diff(start_xp[idx:idx+2]),2)+np.power(np.diff(start_yp[idx:idx+2]),2))) + interp_range = np.linspace(0,1, int(1+section_len/delta)) fx=interp1d(np.linspace(0,1,2),start_xp[idx:idx+2],kind=1) fy=interp1d(np.linspace(0,1,2),start_yp[idx:idx+2],kind=1) @@ -54,9 +53,9 @@ def road_curve(state,track): POLY_RANK = 3 #given vehicle pos find lookahead waypoints - nn_idx=get_nn_idx(state,track)-1 - LOOKAHED = POLY_RANK + 1 - lk_wp=track[:,nn_idx:nn_idx+LOOKAHED] + nn_idx=get_nn_idx(state,track) + LOOKAHED = POLY_RANK*2 + lk_wp=track[:,max(0,nn_idx-1):nn_idx+LOOKAHED] #trasform lookahead waypoints to vehicle ref frame dx = lk_wp[0,:] - state[0] @@ -68,17 +67,32 @@ def road_curve(state,track): #fit poly return np.polyfit(wp_vehicle_frame[0,:], wp_vehicle_frame[1,:], POLY_RANK, rcond=None, full=False, w=None, cov=False) -def f(x,coeff): - """ - """ - return round(coeff[0]*x**3 + coeff[1]*x**2 + coeff[2]*x**1 + coeff[3]*x**0,6) +# def f(x,coeff): +# """ +# """ +# return round(coeff[0]*x**3 + coeff[1]*x**2 + coeff[2]*x**1 + coeff[3]*x**0,6) # def f(x,coeff): # return round(coeff[0]*x**5+coeff[1]*x**4+coeff[2]*x**3+coeff[3]*x**2+coeff[4]*x**1+coeff[5]*x**0,6) -def df(x,coeff): - """ - """ - return round(3*coeff[0]*x**2 + 2*coeff[1]*x**1 + coeff[2]*x**0,6) +def f(x,coeff): + y=0 + j=len(coeff) + for k in range(j): + y += coeff[k]*x**(j-k-1) + return round(y,6) + +# def df(x,coeff): +# """ +# """ +# return round(3*coeff[0]*x**2 + 2*coeff[1]*x**1 + coeff[2]*x**0,6) + # def df(x,coeff): # return round(5*coeff[0]*x**4 + 4*coeff[1]*x**3 +3*coeff[2]*x**2 + 2*coeff[3]*x**1 + coeff[4]*x**0,6) + +def df(x,coeff): + y=0 + j=len(coeff) + for k in range(j-1): + y += (j-k-1)*coeff[k]*x**(j-k-2) + return round(y,6) diff --git a/notebooks/equations.ipynb b/notebooks/equations.ipynb index 4047e11..dec62be 100644 --- a/notebooks/equations.ipynb +++ b/notebooks/equations.ipynb @@ -11,7 +11,7 @@ }, { "cell_type": "code", - "execution_count": 2, + "execution_count": 1, "metadata": {}, "outputs": [ { @@ -28,7 +28,7 @@ "[0, 0, 0, v*cos(psi), 0]])" ] }, - "execution_count": 2, + "execution_count": 1, "metadata": {}, "output_type": "execute_result" } @@ -52,7 +52,7 @@ }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 2, "metadata": {}, "outputs": [ { @@ -69,7 +69,7 @@ "[ sin(psi), 0]])" ] }, - "execution_count": 3, + "execution_count": 2, "metadata": {}, "output_type": "execute_result" } @@ -83,7 +83,7 @@ }, { "cell_type": "code", - "execution_count": 4, + "execution_count": 3, "metadata": {}, "outputs": [ { @@ -98,7 +98,7 @@ "[0, 0, 1]])" ] }, - "execution_count": 4, + "execution_count": 3, "metadata": {}, "output_type": "execute_result" } @@ -120,7 +120,7 @@ }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 4, "metadata": {}, "outputs": [ { @@ -135,7 +135,7 @@ "[ 0, dt]])" ] }, - "execution_count": 5, + "execution_count": 4, "metadata": {}, "output_type": "execute_result" } @@ -156,7 +156,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 5, "metadata": {}, "outputs": [], "source": [ @@ -180,7 +180,7 @@ }, { "cell_type": "code", - "execution_count": 15, + "execution_count": 6, "metadata": {}, "outputs": [ { @@ -196,7 +196,7 @@ "[0, v*(tan(delta)**2 + 1)/L]])" ] }, - "execution_count": 15, + "execution_count": 6, "metadata": {}, "output_type": "execute_result" } @@ -207,7 +207,7 @@ }, { "cell_type": "code", - "execution_count": 10, + "execution_count": 7, "metadata": {}, "outputs": [ { @@ -223,7 +223,7 @@ "[0, 0, dt*tan(delta)/L, 1]])" ] }, - "execution_count": 10, + "execution_count": 7, "metadata": {}, "output_type": "execute_result" } @@ -236,7 +236,7 @@ }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 8, "metadata": {}, "outputs": [ { @@ -252,7 +252,7 @@ "[ 0, dt*v*(tan(delta)**2 + 1)/L]])" ] }, - "execution_count": 11, + "execution_count": 8, "metadata": {}, "output_type": "execute_result" } @@ -263,7 +263,7 @@ }, { "cell_type": "code", - "execution_count": 13, + "execution_count": 9, "metadata": {}, "outputs": [ { @@ -279,7 +279,7 @@ "[-delta*dt*v*(tan(delta)**2 + 1)/L]])" ] }, - "execution_count": 13, + "execution_count": 9, "metadata": {}, "output_type": "execute_result" } @@ -297,7 +297,7 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 10, "metadata": {}, "outputs": [], "source": [ @@ -350,9 +350,18 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 43, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "/home/marcello/miniconda3/envs/jupyter/lib/python3.8/site-packages/IPython/core/interactiveshell.py:3331: RankWarning: Polyfit may be poorly conditioned\n", + " exec(code_obj, self.user_global_ns, self.user_ns)\n" + ] + } + ], "source": [ "#define track\n", "wp=np.array([0,5,6,10,11,15, 0,0,2,2,0,4]).reshape(2,-1)\n", @@ -379,23 +388,62 @@ "#def f(x,coeff):\n", "# return coeff[0]*x**3+coeff[1]*x**2+coeff[2]*x**1+coeff[3]*x**0\n", "def f(x,coeff):\n", - " return coeff[0]*x**5+coeff[1]*x**4+coeff[2]*x**3+coeff[3]*x**2+coeff[4]*x**1+coeff[5]*x**0" + " return coeff[0]*x**5+coeff[1]*x**4+coeff[2]*x**3+coeff[3]*x**2+coeff[4]*x**1+coeff[5]*x**0\n", + "\n", + "def f(x,coeff):\n", + " y=0\n", + " j=len(coeff)\n", + " for k in range(j):\n", + " y += coeff[k]*x**(j-k-1)\n", + " return y\n", + "\n", + "# def df(x,coeff):\n", + "# return round(3*coeff[0]*x**2 + 2*coeff[1]*x**1 + coeff[2]*x**0,6)\n", + "def df(x,coeff):\n", + " y=0\n", + " j=len(coeff)\n", + " for k in range(j-1):\n", + " y += (j-k-1)*coeff[k]*x**(j-k-2)\n", + " return y" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 44, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "text/plain": [ + "array([ 0.10275887, 0.03660033, -0.21750601, 0.03551043, -0.53861442,\n", + " -0.58083993])" + ] + }, + "execution_count": 44, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "coeff" ] }, { "cell_type": "code", - "execution_count": null, + "execution_count": 40, "metadata": {}, - "outputs": [], + "outputs": [ + { + "data": { + "image/png": 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\n", + "text/plain": [ + "

    " + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], "source": [ "import matplotlib.pyplot as plt\n", "plt.style.use(\"ggplot\")\n", @@ -423,6 +471,71 @@ "#plt.savefig(\"fitted_poly\")" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## With BSLINES" + ] + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "(array([-0.39433757, -0.39433757, -0.39433757, -0.39433757, 0.56791288,\n", + " 1.04903811, 1.67104657, 1.67104657, 1.67104657, 1.67104657]), array([-0.34967937, 0.15467936, -2.19173016, 1.11089663, -8. ,\n", + " -0.7723291 , 0. , 0. , 0. , 0. ]), 3)\n", + "[[ 4.64353595 4.64353595 4.64353595 4.64353595 -23.21767974\n", + " 65.74806776 65.74806776 65.74806776 65.74806776]\n", + " [ -6.70236682 -6.70236682 -6.70236682 -6.70236682 6.70236682\n", + " -26.8094673 95.8780974 95.8780974 95.8780974 ]\n", + " [ 1.57243489 1.57243489 1.57243489 1.57243489 1.57243489\n", + " -8.10159833 34.85967446 34.85967446 34.85967446]\n", + " [ -0.34967937 -0.34967937 -0.34967937 -0.34967937 -0.90523492\n", + " -1.1830127 -0.7723291 -0.7723291 -0.7723291 ]]\n" + ] + } + ], + "source": [ + "#define track\n", + "wp=np.array([0,5,6,10,11,15, 0,0,2,2,0,4]).reshape(2,-1)\n", + "track = compute_path_from_wp(wp[0,:],wp[1,:],step=0.5)\n", + "\n", + "#vehicle state\n", + "state=[3.5,0.5,np.radians(30)]\n", + "\n", + "#given vehicle pos find lookahead waypoints\n", + "nn_idx=get_nn_idx(state,track)-1 #index ox closest wp, take the previous to have a straighter line\n", + "LOOKAHED=6\n", + "lk_wp=track[:,nn_idx:nn_idx+LOOKAHED]\n", + "\n", + "#trasform lookahead waypoints to vehicle ref frame\n", + "dx = lk_wp[0,:] - state[0]\n", + "dy = lk_wp[1,:] - state[1]\n", + "\n", + "wp_vehicle_frame = np.vstack(( dx * np.cos(-state[2]) - dy * np.sin(-state[2]),\n", + " dy * np.cos(-state[2]) + dx * np.sin(-state[2]) ))\n", + "\n", + "#fit poly\n", + "import scipy\n", + "from scipy.interpolate import BSpline\n", + "from scipy.interpolate import PPoly,splrep\n", + "spl=splrep(wp_vehicle_frame[0,:], wp_vehicle_frame[1,:])\n", + "coeff\n", + "print( spl)\n", + "print(PPoly.from_spline(spl).c)\n", + "#coeff=np.polyfit(wp_vehicle_frame[0,:], wp_vehicle_frame[1,:], 5, rcond=None, full=False, w=None, cov=False)\n", + "\n", + "#def f(x,coeff):\n", + "# return coeff[0]*x**3+coeff[1]*x**2+coeff[2]*x**1+coeff[3]*x**0\n", + "\n" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -464,14 +577,25 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 45, "metadata": {}, - "outputs": [], + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "-0.5808399313875324\n", + "28.307545725691345\n" + ] + } + ], "source": [ "#for 0\n", "\n", - "cte=coeff[3]\n", - "epsi=-np.arctan(coeff[2])\n", + "# cte=coeff[3]\n", + "# epsi=-np.arctan(coeff[2])\n", + "cte=f(0,coeff)\n", + "epsi=-np.arctan(df(0,coeff))\n", "print(cte)\n", "print(np.degrees(epsi))" ]

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