## 3 Results
### Estimation of Load Weight  
This section presents a methodology for estimating the transported load weight in an electric forklift based on the electrical power measured at the lifting motor. The proposed approach leverages the relationships previously established between lifting power, lifting efficiency, and load weight. The estimation process follows these steps:  
1. Measurement of lifting power and energy: The instantaneous power at the lifting motor is recorded during the lifting cycle. From this signal, three important variables are extracted: the electrical lifting energy (\(E_{el}\)), the median electric lifting power (\(P_{el-med}\)), and the average lifting speed (\(V_{l-avg}\)). These metrics are required inputs for the following stages. 2. Estimation of lifting efficiency: The lifting efficiency is estimated using either the linear or the multivariable regression model derived in the previous section. In the linear case, efficiency is expressed as a function of the median lifting power only. In contrast, the multivariable model incorporates both the median lifting power and the average lifting speed to capture the combined effect of electrical and kinematic variables. When the linear approximation is used, the following model is applied: \[\hat{\eta_{l}}=a\cdot P_{el-med}+b\] (6) where \(a\) and \(b\) are regression coefficients obtained from the empirical data. In this case \(a=2.77\ b=-3.92\). In the case of the multivariable approximation, the following model is applied: \[\hat{\eta_{l}}=a_{0}+a_{1}P_{el-med}+a_{2}V_{l-avg}+a_{3}P_{el-med}^{2}+a_{4} P_{el-med}V_{l-avg}+a_{5}V_{l-avg}^{2}\] (7) where Table 3 shows the numerical values of the regression coefficients \(a_{n}\). 3. Estimation of potential energy: The estimated potential energy (\(\hat{E}_{pl}\)) is then computed by rearranging the definition of lifting efficiency: \[\hat{E}_{pl}=\hat{\eta_{l}}\cdot E_{el}\] (8) where \(E_{el}\) is the electrical energy measured during the lifting cycle (step 1). Figure 10: Dependence of lifting efficiency on \(P_{el-med}\) and \(V_{l-avg}\). Estimation of transported mass: From the estimated potential energy, the estimated transported mass (\(\hat{m}\)) is derived using the gravitational energy equation:  
\[\hat{m}=\frac{\hat{E}_{pl}}{g\cdot h} \tag{9}\]  
where \(h\) is the fixed lifting height used in step 1 (\(h=2\) m in this case). Initially, a linear approximation model was used to estimate the lifting efficiency as a function of the median electrical power. Figure (a)a presents a comparison between the actual transported mass and the estimated mass for each lifting event indexed from 0 to 53, indicating that the estimated values closely follow the real data across various load conditions. Figure (b)b displays the percentage error for each test case, revealing a generally consistent performance. The results confirm that the methodology is effective for indirect estimation of the transported load using only electrical power signals. The maximum error reached nearly +26%, indicating overestimation for a load weight of 1202 kg, while the minimum estimation error observed was approximately \(-0.23\%\) (underestimation) for a real load of 500.5 kg. The mean absolute percentage error (MAPE) was 10.6%, demonstrating that the method provides sufficiently accurate results for practical applications, without requiring physical load sensors. The MAPE is calculated as follows:  
\[\text{MAPE}=\frac{100}{n}\sum_{i=1}^{n}\left|\frac{\hat{m}_{i}-m_{i}}{m_{i}}\right| \tag{10}\]  
where \(\hat{m}_{i}\) is the estimated mass, \(m_{i}\) is the actual mass, and \(n\) is the total number of lifting events evaluated. While the single-variable approach yielded satisfactory results, its accuracy was limited by the inherent variability of the lifting conditions. To address this issue, a second, more accurate model was developed using a multivariable regression that incorporates both the median electrical power and the average lifting speed.