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In the process of using computers to solve scientific calculation problems, four types of errors must be considered: model errors, observation errors, truncation errors (or method errors), and rounding errors .
1. Model error
To solve scientific computing problems with computers, we must first establish a mathematical model, which is obtained by abstracting and simplifying the actual problem described, so it is approximate. We call the error between the mathematical model and the actual problem the model error .
For example, when identifying robot dynamics parameters, the friction force/torque model of robot joints is usually approximated as a first-order model: τ = fcsign (q ˙) + fvq ˙ \tau=f_csign(\dot{q})+f_v\dot {q}τ=fcsign(q˙)+fvq˙After linearizing the dynamic equation, the entire dynamic parameter identification model is a linear least squares model, which greatly simplifies the difficulty of solving and improves the robustness of the model. In fact, at low speeds, there is a non-linear relationship between joint friction/moment and joint speed.
2. Observation error
In the mathematical model, there are often some physical quantities obtained based on observations, and these physical quantities obviously also contain errors. This kind of error caused by observation is called observation error .
3. Truncation error (or method error)
In many cases, it is difficult to obtain an accurate solution to a mathematical model established from actual problems. Numerical methods are usually used to find its approximate solution. For example, the infinite calculation process is often replaced by a finite calculation process. The error between the exact solution and the approximate solution obtained by numerical methods is called truncation error . Because the truncation error is inherent to the numerical calculation method, it is also called method error .
For example, using Taylor polynomial P n (x) = f (0) + f ′ (0) 1! X + + f ′ ′ (0) 2! X 2 +... + Fn (0) n! Xn P_n(x )=f(0)+\frac{f'(0)}{1!}x + +\frac{f''(0)}{2!}x^2+...+\frac{f^ {n}(0)}{n!}x^nPn(x)=f(0)+1!f′(0)x++2!f′′(0)x2+...+n!fn(0)xWhen n is approximately replaced, the truncation error of the numerical method isR n (x) = f (x) − P n (x) = fn + 1 (ξ) (n + 1)! xn + 1 R_n(x)=f( x)-P_n(x)=\frac{f^{n+1}(\xi)}{(n+1)!}x^{n+1}Rn(x)=f(x)−Pn(x)=(n+1)!fn + 1 (p)xn + 1，ξ \ xiξ inxxx and0 0Between 0 .
4. Rounding error
When calculating with a computer, due to the limited word length of the computer, the original data will produce errors on the computer. The calculation process may produce new errors. This error is called rounding error . For example, 3.14159 is used to approximate π \piπ , the resulting errorR = π − 3.14159 = 0.0000026… R=\pi-3.14159=0.0000026\cdotsR=Pi−3.14159=0.0000026⋯
references
Numerical Analysis (4th Edition) Li Qingyang, Wang Nengchao, Yi Dayi