diff --git a/11-Extended-Kalman-Filters.ipynb b/11-Extended-Kalman-Filters.ipynb index 6082f0c..5de983f 100644 --- a/11-Extended-Kalman-Filters.ipynb +++ b/11-Extended-Kalman-Filters.ipynb @@ -964,15 +964,15 @@ "We find The $\\mathbf F$ by taking the Jacobian of $f(x,u)$.\n", "\n", "$$\\mathbf F = \\frac{\\partial f(x, u)}{\\partial x} =\\begin{bmatrix}\n", - "\\frac{\\partial \\dot x}{\\partial x} & \n", - "\\frac{\\partial \\dot x}{\\partial y} &\n", - "\\frac{\\partial \\dot x}{\\partial \\theta}\\\\\n", - "\\frac{\\partial \\dot y}{\\partial x} & \n", - "\\frac{\\partial \\dot y}{\\partial y} &\n", - "\\frac{\\partial \\dot y}{\\partial \\theta} \\\\\n", - "\\frac{\\partial \\dot{\\theta}}{\\partial x} & \n", - "\\frac{\\partial \\dot{\\theta}}{\\partial y} &\n", - "\\frac{\\partial \\dot{\\theta}}{\\partial \\theta}\n", + "\\frac{\\partial f_1}{\\partial x} & \n", + "\\frac{\\partial f_1}{\\partial y} &\n", + "\\frac{\\partial f_1}{\\partial \\theta}\\\\\n", + "\\frac{\\partial f_2}{\\partial x} & \n", + "\\frac{\\partial f_2}{\\partial y} &\n", + "\\frac{\\partial f_2}{\\partial \\theta} \\\\\n", + "\\frac{\\partial f_3}{\\partial x} & \n", + "\\frac{\\partial f_3}{\\partial y} &\n", + "\\frac{\\partial f_3}{\\partial \\theta}\n", "\\end{bmatrix}\n", "$$\n", "\n", @@ -1087,9 +1087,9 @@ "If this was a linear problem we would convert from control space to state space using the by now familiar $\\mathbf{FMF}^\\mathsf T$ form. Since our motion model is nonlinear we do not try to find a closed form solution to this, but instead linearize it with a Jacobian which we will name $\\mathbf{V}$. \n", "\n", "$$\\mathbf{V} = \\frac{\\partial f(x, u)}{\\partial u} \\begin{bmatrix}\n", - "\\frac{\\partial \\dot x}{\\partial v} & \\frac{\\partial \\dot x}{\\partial \\alpha} \\\\\n", - "\\frac{\\partial \\dot y}{\\partial v} & \\frac{\\partial \\dot y}{\\partial \\alpha} \\\\\n", - "\\frac{\\partial \\dot{\\theta}}{\\partial v} & \\frac{\\partial \\dot{\\theta}}{\\partial \\alpha}\n", + "\\frac{\\partial f_1}{\\partial v} & \\frac{\\partial f_1}{\\partial \\alpha} \\\\\n", + "\\frac{\\partial f_2}{\\partial v} & \\frac{\\partial f_2}{\\partial \\alpha} \\\\\n", + "\\frac{\\partial f_3}{\\partial v} & \\frac{\\partial f_3}{\\partial \\alpha}\n", "\\end{bmatrix}$$\n", "\n", "These partial derivatives become very difficult to work with. Let's compute them with SymPy. "